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of  the 

University  of  California 


Los  Angeles 


Form  L  1 


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JAR  2  9  1329 

JAN  24  tg 
MAY-lg  1933 

3  1991 


Form  L-9-15jn-8,'26 


The  Teaching  of  Mathematics 


The  Teaching  of 


MATHEMATICS 


BY 


RAYMOND  E.  MANCHESTER,  A.M. 

PROFESSOR  OF  MATHEMATICS,  STATE  NORMAL  SCHOOL 
OSHKOSH,  Wis. 


SYRACUSE,  N.  Y.< 
C.  W.  BARDEEN,   PUBLISHER 

1913 


COPYRIGHT,  1913,  BY  RAYMOND  E.  MANCHESTER 


OA\\ 
M   3 


Co 


CONTENTS 

PAGE 
A  period  of  reconstruction 9 

Meaning  of  education 15 

Values  of  mathematics 18 

Daily  use 23 

Three  groups  of  facts 25 

Habit  of  logical  thought 30 

Generalizing  power 33 

Power  to  grasp  situations 35 

Individual  thinking 37 

Reconstruction  of  arithmetic 39 

Materials  of  mathematics 41 

Arithmetic  by  grades 44 

Algebra 53 

Geometry 54 

Vocational  departments 56 

Methods  and  Modes 58 

Analysis  and  synthesis 61 

Induction  and  deduction 63 

The  Heuristic  method 64 

(7) 


8  CONTENTS 

The  laboratory  method 65 

The  Socratic  method . .-. 66 

The  Montessori  method 67 

Interest  essential 69 

Lecture  and  recitation  modes 71 

Summary 72 


Mathematics  and  Education 


We  are  in  a  state  of  transition.  The  ideals  of 
civilization  are  changing.  Never  in  the  history  of 
mankind  has  the  condition  of  unrest  been  so  uni- 
versal. It  is  a  dominant  force.  Whether  or  not 
the  transition  be  from  a  lower  to  a  higher  plane 
depends  much  upon  the  teacher.  One  need  not 
be  a  philosopher  to  appreciate  this.  Even  those 
who  think,  work,  and  live  by  the  day  are  capable 
of  understanding  the  situation.  During  the  span 
of  the  last  one  hundred  years  the  human  race  has 
brought  to  a  conclusion  movements  three  thousand 
years  in  the  forming.  When  one  pauses  long  enough 
to  reflect,  truly  marvelous  are  the  strides  mankind 
has  taken.  The  whole  method  and  mode  of  living 
have  changed.  With  steam,  oil,  electricity,  steel, 
cement,  and  light  and  sound  waves  man  has  elevated 
himself  to  a  place  only  given  the  gods  in  times  re- 
mote. Not  only  is  he  despotic  ruler  of  the  earth, 
but  he  reaches  out  his  long,  powerful,  scientific  arms 
to  grasp  the  secrets  of  the  universe.  He  has  ex- 
tended his  vision  in  two  directions,  to  those  regions 
his  eye  was  not  created  to  gaze  upon.  With  the 
telescope  he  looks  into  God's  own  workshop  and 
sees  the  stars  and  planets  in  the  making;  with  the 
microscope  he  searches  out  the  infinitesimal  parts 

(9) 


10  MATHEMATICS  AND  EDUCATION 

of  His  universe.  His  analysis  is  carried  even  unto 
himself,  so  that  his  own  body,  mind  and  soul  are 
known  in  their  parts  and  relationships.  He  even 
finds  an  analysis  of  the  analysis. 

Man  is  uprooting  all  that  was  considered  constant ; 
upheaving  the  very  foundations  of  the  old  regime; 
making  over  his  politics,  his  ethics,  his  morals,  yes 
even  his  religion.  Is  it  any  wonder  that  he  is  dis- 
satisfied with  his  educational  system?  Is  it  any 
wonder  that  all  those  things,  which  appeared  sane 
to  his  forbears,  seem  to  his  frenzied  mind  insane? 
Is  it  strange  that  the  world  is  filled  with  promoters, 
agitators,  reformers,  and  purists?  Is  it  to  be  won- 
dered at  that  sophistry  finds  ideal  conditions  for 
germination  ? 

By  no  means  is  educational  unrest  out  of  place. 
It  is  the  necessary  product  of  the  situation.  There 
could  be  no  other  result.  Educational  unrest, 
though  grating  to  some,  is  nevertheless  the  logical 
thing.  To  close  our  eyes  to  it  is  folly,  and  is  equiva- 
lent to  dying  with  the  old  system.  Because  of  this 
prevailing  state  of  unrest  two  questions  have  shaped 
themselves:  the  one,  what  shall  we  offer  the  child 
who  is  destined  by  social  law  to  serve  as  a  worker, 
and,  the  other,  by  what  method  shall  the  material 
of  this  instruction  be  presented?  In  answer  to 
these  questions  two  great  movements  stand  out 
distinctly,  the  one  toward  vocational  education, 
the  other  toward  informality.  With  reference  to 
the  first  movement,  educators  at  last  agree  that 
public  education  should  face  the  96%  who  are  to 


RECONSTRUCTION  11 

fill  the  producing  class,  the  so-called  masses.  With 
regard  to  the  second  movement,  new  methods  are 
found  to  be  more  efficient  than  the  old,  because 
men  and  women  at  last  have  discovered  that  the 
schoolroom  contains  red-blooded  children. 

Perhaps  no  subject  is  so  directly  in  the  flux  of 
the  instability  of  things  as  mathematics.  For  cen- 
turies this  subject  has  held  the  unenviable  reputa- 
tion of  being  tortuous;  yet  man,  because  of  his  un- 
ending need  of  the  subject,  has  endured  silently. 
It  has  been  one  of  the  necessary  evils  of  life,  yet, 
did  man  but  know  it,  wonderfully  full  of  beauty  in 
its  symmetry  and  exactness.  Indispensible  to  the 
man  who  builds,  to  the  man  who  barters,  to  the 
man  who  meditates,  it  has  ever  been  shrouded  with 
a  mantle  of  mysticism.  One  might  say  with  truth 
that  they  who  have  the  greatest  need  of  the  find- 
ings of  mathematical  investigation  have  the  smallest 
opportunity  to  acquire  a  knowledge  of  them. 

The  fundamental  principles  of  mathematics  could 
be  written  upon  a  calling  card.  From  these  simple 
facts  a  vast  amount  of  detail  has  been  accumulated 
through  almost  endless  deductions.  Under  the 
spell  of  genius  there  has  been  constructed  a  most 
wonderful  edifice  upon  this  foundation,  a  struct- 
ure similar  in  many  ways  to  a  marvelous  mansion 
with  exquisite  detail  beautiful  to  look  upon,  but 
next  to  useless  to  the  man  who  is  to  have  a  home. 
In  both  cases  a  wonderful  foundation  supports  a 
structure  of  luxury,  when  by  careful  construction, 
a  useful  as  well  as  an  ornamental  structure  might 
have  been  built. 


12  MATHEMATICS  AND  EDUCATION 

The  magnificent  is  highly  in  order  when  the  needs 
of  mankind  have  been  satisfied.  With  the  96% 
asking  for  recognition  it  is  only  proper  that  the 
first  thought  should  be  toward  utility.  So  in  the 
reconstruction  of  mathematics  the  utilitarian  aspect 
is  first  in  order,  a  mathematics  for  the  builder  and 
the  barterer.  To  present  such  material  in  an  in- 
teresting manner  is  an  approximation  to  our  ideal. 
To  humanize  mathematics  is  the  problem.  The 
solution  does  not  lie  merely  in  changing  the  order 
of  presentation  or  in  shortening  the  course  by  re- 
moving certain  subjects,  as  many  reformers  firmly 
believe,  judging  by  the  many  and  varied  text-books 
on  the  market.  The  teacher  must  take  the  re- 
sponsibility of  the  task  and  discover  the  secret  of 
life.  To  do  this,  informality  is  absolutely  necessary. 

To  speak  of  the  need  of  thorough  training  before 
an  attempt  is  made  to  teach  mathematics  would 
seem  unnecessary  were  it  not  for  the  fact  that  in- 
competence is  the  rule  rather  than  the  exception. 
Only  in  recent  years  under  the  guidance  of  general 
pedagogy  have  members  of  the  teaching  profession 
come  to  the  realization  that  the  difficulty  of  the 
subject  calls  for  special  preparation.  The  great 
number  of  under-prepared  teachers  in  the  profes- 
sion has  caused  a  false  philosophy  to  grow  up ;  even 
in  our  modern  renaissance  the  bug-bear  of  practi- 
cability has  caused  an  addition  to  the  confusion. 
It  has  come  to  be  recognized  as  a  truth  that  there 
is  an  art  of  teaching  a  science.  Teachers  are  be- 
ginning to  appreciate  the  fact  that  mere  acquain- 


STUDY  OF  NUMBER  13 

tance  with  the  science  is  not  a  sufficient  preparation 
for  the  teaching  of  it. 

While  this  unrest  has  found  most  pronounced  ex- 
pression in  the  new  methods  of  teaching  the  lin- 
guistic subjects,  there  have  been  many,  more  or 
less  successful,  attempts  to  remedy  existing  evils 
in  the  presentation  of  the  subjects  relating  to  the 
development  of  the  number  concept.  Especially 
with  regard  to  elementary  arithmetic  is  this  true. 
Even  yet,  however,  aptness  in  "riggers"  is  deemed 
sufficient  qualification  for  the  teacher  in  mathematics 
in  many  schools.  Thanks,  however,  to  the  develop- 
ment of  the  psychological  and  pedagogical  view- 
points, this  custom  is  rapidly  falling  into  disrepute. 

In  apprehending  the  objective  world,  the  indi- 
vidual is  quite  as  appreciative  of  the  number  rela- 
tion as  any  other.  It  is  a  fundamental  truth  that 
he  is  conscious  of  his  world  in  number  as  well  as 
in  time  and  space.  Whether  this  relation  be  co- 
ordinate or  subordinate  need  not  greatly  concern 
us  except  in  philosophic  discussion.  The  mere 
fact  of  its  existence  makes  necessary  a  reaction  on 
our  part.  In  our  educative  system  it  is  essential 
that  attention  be  given  to  the  study  of  the  number 
phenomena,  and  that  methods  be  devised  for  the 
study  which  shall  approximate  the  best.  Through- 
out all  the  centuries  man  has  studied  number,  in- 
venting machinery  in  the  form  of  symbol  systems 
for  his  experiments.  And,  if  we  are  to  judge  fairly 
the  rise  of  civilization,  it  is  quite  as  dependent  upon 
the  mechanical  arts  as  upon  the  growth  of  language. 


14  MATHEMATICS  AND  EDUCATION 

All  the  mechanical  arts  are  dependent  upon  the 
basic  facts  of  number.  Yes,  even  the  machinery 
of  language  is  built  upon  the  supposition  that  these 
relations  are  valid. 

The  prevailing  weakness  of  the  mathematics 
teacher  has  led  to  two  results,  the  appreciation  of 
the  fact  that  the  subject  is  one  of  the  hardest  to 
teach,  and  the  rise  of  the  question  of  revising  the 
course  as  offered.  The  pedagogy  of  mathematics 
is  receiving  world- wide  consideration.  In  the  older 
countries  this  is  especially  true,  owing,  no  doubt,  to 
the  superior  treatment  of  the  science  of  general 
pedagogy.  In  America,  although  the  problem  is 
not  so  nearly  solved,  it  is  equally  as  vital  and  should 
call  forth  the  best  talent  available.  The  findings 
of  the  many  societies,  the  discussions  in  scientific 
publications,  the  founding  of  departments  of  educa- 
tion in  our  universities,  and  the  reform  of  our  nor- 
mal schools  have  been  the  dominant  factors  in  the 
revolution  in  America.  It  is  to  be  hoped  that  the 
mathematics  teacher  of  the  future  may  be  so  pre- 
pared that  he  or  she  can  overcome  the  temptation 
to  teach  by  rote  and  to  follow  time  worn  customs. 
The  prevailing  unrest  is  evidence  of  thought.  Per- 
haps the  movement  toward  the  practical  will  over- 
carry — in  all  probability  it  will — but  we  may  be 
thankful  that  there  is  movement.  The  subject 
is  as  thoroughly  alive  as  any  in  the  curriculum  if  the 
chains  of  bondage  are  loosened.  Let  us  pray  that 
the  readjustment  be  rapid. 


MEANING  OF  EDUCATION  15 

To  justify  a  place  for  the  study  of  mathematics 
as  a  course  of  study  in  our  educational  system,  it  is 
first  necessary  to  have  a  definite  idea  of  the  mean- 
ing of  education.  We  must  have  a  solid  base  for 
that  which  becomes  the  superstructure.  This  is 
not  difficult,  inasmuch  as  the  basic  principles  are 
so  simple  as  to  be  accepted  by  all. 

A  simple  analysis  is  to  consider  society  held  in- 
tact through  three  relationships,  (a)  the  relation- 
ship of  the  individual  to  the  group,  (6)  the  relation- 
ship of  the  group  to  the  individual,  (c)  the  relation- 
ships of  the  individual  and  of  the  group  to  nature. 
The  relationships  in  turn  might  be  subdivided  into 
other  relationships,  but  in  this  brief  discussion 
the  three  mentioned  are  sufficient.  It  being  evident 
that  individuality  has  no  meaning  apart  from  the 
group,  that  the  group  has  no  meaning  apart  from 
the  individual,  and  that  nature  has  no  meaning 
except  in  relation  to  the  individual  action  and  group 
action,  we  may  be  safe  in  considering  education  to 
have  meaning  only  when  the  relationships  exist. 
Education,  whether  it  refers  to  the  individual  or 
to  the  group,  has  for  its  aim  the  perfection  of  rela- 
tionship. Education  for  the  individual  must  have 
for  its  object  the  individual's  better  reaction  to  the 
group  and  to  nature;  education  for  the  group,  must 
have  for  its  object  the  groupbetter  reaction  to  the 
individual  and  to  nature. 

Thus  it  is,  that  attempts  to  state  specifically  what 
education  is  are  so  futile,  there  being  such  a  multi- 
tude of  minor  relationships  and  reactions  involved 


16  MATHEMATICS  AND  EDUCATION 

in  the  three  larger  relationships.  Certain  it  is,  that 
education  is  a  process  that  must  constantly  vary. 
Keeping  these  facts  in  mind,  it  is  not  difficult  for 
one  to  appreciate  the  fact  that  education  is  intimate- 
ly connected  with  the  conscious  aim  of  the  indi- 
vidual and  of  society  Broadly,  education  is  at- 
tained only  after  a  life  has  been  lived  through,  only 
after  the  individual  has  drawn  from  experience  in 
reaction  of  every  sort;  but  for  the  teacher,  education 
refers  to  that  elementary  beginning  found  in  the 
school.  Education  is  the  acquisition  of  power  to 
react  toward  the  approximation  of  certain  ideals, 
and  of  power  to  organize  such  reactions.  The  as- 
pects of  education  are  many  and  varied,  as  for 
example,  the  biological,  the  psychological,  the 
physiological,  the  ethical,  the  historical,  and  the 
philosophic,  but  back  of  all  is  the  fundamental 
principle  that  education  becomes  meaningful  only 
with  regard  to  action  and  reaction. 

It  follows  that  education  does  not  have  the  same 
meaning  for  all.  Group  ideals  call  for  differing 
reactions.  For  those  following  trade  life,  quite 
different  reactions  are  called  for  than  for  those  fol- 
lowing professional  life.  School  training  must  be 
carried  on  with  due  appreciation  of  this  fact.  The 
materials  selected  and  the  method  adopted  in  pre- 
sentation should  be  such  as  the  group  ideal  may 
demand. 

There  is,  of  course,  a  certain  constant  of  educa- 
tion based  upon  the  common  characteristics  of  all 
men.  This  constant  is  well  understood.  It  is  the 


POWER  TO  SERVE  17 

training  of  proper  reactions  to  organized  society, 
as  such,  with  reference  to  the  home,  the  state,  and 
the  church.  Always,  the  point  of  contact  is  in  the 
life  of  the  individual.  Defined  in  these  terms  edu- 
cation is  power  to  react.  Education  is  power  to 
serve.  To  become  educated  is  to  attain  such  an 
approximation  to  the  highest  self  as  to  be  able  to 
serve  well.  Self-perfection,  whether  it  be  in  the 
individual  life  or  in  the  group  life,  is  the  most  ef- 
fective method  of  giving  service.  The  individual 
at  best  is  a  unit  in  the  group,  and  as  such,  serves 
best  as  an  approximately  perfect  unit;  while,  on 
the  other  hand,  the  group  is  a  totality  of  units 
having  individual  unity  determined  by  the  strength 
of  its  parts. 

Surely  then,  if  power  is  education,  the  materials 
of  education  must  be  those  best  suited  to  produce 
power,  and  the  methods  of  education  must  be  those 
best  adapted  to  the  turning  of  these  materials  into 
power.  In  the  selection  of  subjects  for  study,  those 
richest  in  power-producing  material  must  be  chosen, 
while  those  most  barren  must  be  left  out.  If  the 
subject  of  mathematics  is  to  stand  such  a  test,  it 
must  show  that  its  material  is  that  which  can  be 
used  in  the  production  of  such  a  power,  and  that  a 
method  of  teaching  is  possible  by  which  this  power 
can  be  generated.  The  successful  teacher  of  math- 
ematics must,  (a)  become  thoroughly  convinced  of 
the  value  of  mathematics  to  the  process  of  educa- 
tion, (6)  make  selection  of  material  best  adapted 
to  the  best  development  of  the  educational  process, 


18  MATHEMATICS  AND  EDUCATION 

(c)   devise  a  best  method  to  present  the  selected 
material  to  the  student. 

Values  of  Mathematics 

The  values  of  mathematics  to  the  individual  and 
to  society  should  be  determined  by  test  of  action 
and  reaction.  It  should  give  to  the  individual, 
greater  ability  to  react  toward  society  to  his  better- 
ment; it  should  give  society,  greater  opportunity 
to  offer  the  individual  conditions  for  a  more  perfect 
life  and  also  greater  efficiency  in  reaction  to  nature. 
Mathematics  is  valuable  to  the  individual  in  offer- 
ing those  methods  and  facts  which  aid  him  in  the 
conduct  of  daily  duties,  and  in  helping  him  to  develop 
power  to  react  to  social  forms,  institutions,  and 
ideals.  Mathematics  is  valuable  to  society  in  of- 
fering material  for  a  closer  knitting  together  of 
institutional  life,  for  the  creation  of  ideals,  and  for 
the  engineering  successes  which  turn  the  infinite 
powers  of  nature  to  the  services  of  mankind. 

With  language,  number  holds  the  center  of  the 
stage  in  educational  inquiry.  To  recognize  only 
the  fact  value  of  the  subject  is  quite  as  narrow  as 
to  recognize  only  the  other  value.  Thus,  to  be  an 
efficient  teacher  of  mathematics,  one  must  acquire 
a  certain  balance.  If  one  were  to  think  the  whole 
of  any  situation  through,  it  would  be  found  that 
the  facts  of  mathematics  are  involved  in  some  way. 
-  All  nature's  laws  are  interpreted  in  number.  Many 
of  the  more  common  phenomena  are  seldom  thought 
of  in  number  relation,  owing  to  the  general  knowledge 
of  the  elements  of  the  subject,  but  all  the  deeper 


VALUES  OF  MATHEMATICS  19 

discussion  is  absolutely  dependent  upon  number 
relation.  '  Such  sciences  as  astronomy  and  land 
measurement  are  almost  entirely  mathematical. 
The  commonness  of  most  of  the  relations  of  weight 
and  measures  is  responsible  for  the  lack  of  appre- 
ciation of  the  mathematical  connections.  In  all 
sciences  mathematics  is  indirectly  connected. 

Also,  a  great  majority  of  the  occupations,  into 
which  students  are  likely  to  go,  require  mathemati- 
cal knowledge  for  successful  participation.  The 
great  rise  of  industrial  occupations  generates  a 
particular  need  of  training.  Common  business 
intercourse  calls  at  least  for  the  elements,  while 
so-called  big-business  is  utterly  dependent  upon 
the  validity  of  number  laws. 

Not  only  has  the  subject  a  great  fact  value  to 
society,  but  it  typifies  a  method  of  reasoning,  accept- 
ed by  society  as  one  of  the  most  effective.  Many 
consider  this  value  far  greater,  directly  and  indi- 
rectly, to  the  student.  All  life  problems  must 
undergo  analysis  before  a  solution  can  be  obtained. 
Situations  must  be  understood  in  the  relationship 
of  their  parts.  To  be  capable  of  handling  a  situa- 
tion is  power,  regardless  of  occupation  or  social  posi- 
tion. The  study  of  mathematics  is  one  of  the  most 
effective  means  to  such-  an  end.  • 

In  considering  the  values  of  mathematics,  one 
is  led  into  consideration  of  the  larger  discussions 
upon  education.  The  awakening  of  the  public  has 
been  stupendous  during  the  past  few  years.  In 
every  state,  city  and  town,  the  awakening  is  taking 


20  MATHEMATICS  AND  EDUCATION 

place.  Teacher  and  parent  alike  are  striving  to 
reconstruct  a  worn-out  system,  and  despite  many 
mistakes  are  accomplishing  their  end.  One  can 
imagine  that  some  huge  giant  is  sending  chains  of 
bondage  flying,  after  fifty  years  of  stupor.  The 
producing  class  of  this  group  of  80,000,000  people 
is  becoming  suddenly  appreciative  of  the  fact  that 
it  has  been  paying  for  something  it  has  not  received, 
and,  with  the  spirit  of  new- world  liberty  fresh  in  its 
blood,  is  striking  for  its  rights.  And  it  will  get 
them. 

The  public,  no  doubt,  has  good  grounds  for  de- 
manding a  change  in  the  educational  system.  In 
fact,  it  has  been  admitted  by  school  men  themselves 
that  the  system  is  not  efficient.  There  has  been, 
and  is,  educational  waste.  The  public  recognizes 
it  in  the  inability  of  school  graduates  to  cope  with 
workaday  problems.  The  public  feels  that  the 
average  graduate  is  not  a  sufficient  reward  for  the 
time,  energy,  and  expense  necessary  for  his  produc- 
tion. In  reaction,  the  public  has  made  severe 
criticism,  verbally  and  in  action.  Through  the 
public  press,  countless  articles  have  been  presented, 
showing  faults  and  offering  suggestions.  In  action, 
much  money  has  been  spent  for  expert  service  and 
for  the  construction  of  excellent  school  buildings 
equipped  with  the  most  modern  apparatus.  If  we 
are  to  judge  by  attempted  reforms,  the  public  has 
arrived  at  three  conclusions  through  its  investiga- 
tion: 1st,  that  too  much  time  is  spent  on  theory 
and  method,  and  too  little  time  devoted  to  facts, — 


FACT  VALUES  OVERESTIMATED  21 

evidenced  by  the  use  of  correspondence  schools, 
technical  schools,  short  course  professional  schools, 
and  by  the  introduction  of  fact-courses  into  the 
public  schools ;  2nd,  that  the  courses  of  study  should 
face  the  masses  more  directly, — evidenced  by  night 
schools,  vocational  departments,  summer  schools, 
and  by  the  cutting  out  of  so-called  culture  courses; 
3rd,  that  the  physical  education  of  the  child  is  equal- 
ly as  important  as  the  mental  education, — evidenced 
by  the  rapid  increase  in  the  number  of  gymnasia 
in  schoolhouses  and  of  teachers  of  physical  educa- 
tion. 

The  first  two  conclusions  vitally  affect  the  teacher 
of  mathematics.  He  must,  or  rather  she  must  — 
there  being  more  women  engaged  in  teaching  than 
men — thoroughly  appreciate  the  need  of  change 
and  be  able  to  make  changes  for  the  best.  With 
regard  to  the  first  conclusion,  there  is  no  doubt  that 
the  fact  values  of  mathematics  are  over  estimated 
and  therefore  over-stressed.  The  public,  in  attempt- 
ing to  remedy  one  evil,  is  very  likely  to  impose 
another.  The  facts  of  mathematics  necessary  to 
everyday  existence  are  only  the  most  elementary 
ones.  Merely  to  cram  these  into  the  students' 
heads,  together  with  many  others  necessary  to  fill 
out  a  course,  is  as  futile  toward  the  giving  of  an 
education  as  cramming  the  dictionary  would  be. 
Observation  without  reflection  is  barren.  Never- 
theless there  is  an  excellent  lesson  to  be  drawn  from 
the  demand  for  facts 


22  MATHEMATICS  AND  EDUCATION 

The  schools  for  the  past  seventy-five  years  have 
been  preparing  students  for  colleges  and  universi- 
ties. The  colleges  and  universities  have  dictated 
the  policy.  Before  the  rise  of  the  public  schools 
to  their  present  importance,  this  system  was  satis- 
factory because  few  students  attended  school  regu- 
larly who  did  not  aspire  to  a  college  education. 
Conditions  have  changed,  however,  without  corre- 
sponding change  in  the  course  of  study.  Now  96% 
of  the  students  in  the  public  schools  do  not  reach 
the  college.  Yet  they  prepare  themselves  for  it. 
The  public  very  justly  is  demanding  a  change.  In 
mathematics  it  demands  a  change  and,  without 
the  opportunity  for  due  reflection,  decides  that 
facts  must  be  emphasized.  Very  good,  but  the 
value  of  mathematics  is  not  alone  in  the  facts.  It 
is  valuable  as  well  in  offering  a  method  of  thought. 
This  is  appreciated  by  the  teacher  and  causes  his 
opposition  to  the  public  demand.  The  public  de- 
mand is  in  fact  just  and  must  be  recognized.  The 
teacher  must  give  and  take.  A  system  of  mathe- 
matics teaching  must  be  established,  giving  the 
facts  demanded  by  the  public  and  also  giving  men- 
tal development.  Such  a  system  will  be  discussed 
later. 

The  valuable  facts  of  mathematics  may  be  grouped 
in  three  classes:  1st,  those  simple  facts  of  use  in  the 
living  out  of  average  daily  life,  which  are  acquired  in 
the  study  of  elementary  mathematics;  2nd,  those 
facts  of  contingent  value;  and  3rd,  those  special 
facts  which  can  be  applied  in  engineering,  in  the 


DAILY  USE  OF  MATHEMATICS  23 

arts,  in  business,  or  in  research.  While  it  is  true 
that  no  subject  outside  of  language  has  such  direct 
application  to  daily  life,  yet  the  importance  to  most 
persons  is  indirect.  But  even  so,  there  is  used  in 
daily  life  only  the  elements  of  any  subject.  How 
often  does  the  man  of  the  street  use  his  knowledge 
of  history,  of  botany,  of  Latin,  of  chemistry,  of 
French  ?  All  people  must  converse  and  all  people 
must  know  enough  of  mathematics  to  carry  on 
ordinary  business.  Beyond  this  the  fact  value  of 
mathematics  affects  the  majority  of  people  indi- 
rectly. Teachers  should  recognize  this  perfectly 
and  drill  eternally  on  elementary  processes,  but 
not  to  the  exclusion  of  the  methods  of  reasoning. 
The  greatest  value  of  the  subject  from  the  stand- 
point of  fact  is  perhaps  not  realized  by  the  great 
majority  of  people.  The  average  citizen  living  out 
his  daily  routine  of  duties  seldom,  if  ever,  takes 
time  to  think  how  vitally  his  life  is  controlled  by 
the  laws  of  number.  From  the  time  he  starts  work 
in  the  morning  until  he  rests  at  night  he  is  constant- 
ly making  use  of  the  practical  application  of  mathe- 
matics. Means  of  communication  and  transporta- 
tion have  been  perfected  until  we  are  truly  citizens 
of  the  world,  getting  world  news  daily,  and  eating 
food  raised  many  hundreds  of  miles  away.  Great 
bridges  make  it  possible  to  cross  waters,  wonderful 
buildings  make  concentration  of  business  possible, 
ocean  lines  connect  the  old  and  new  worlds.  Man 
has  become  cosmopolitan  in  the  superlative.  And 
what  a  wonderful  force  mathematics  has  been,  the 

LOS  ANGELES 

vlAL  cCHOQt 


24  MATHEMATICS  AND  EDUCATION 

foundation  upon  which  this  vast  system  has  been 
built. 

From  the  historical  viewpoint,  the  fact  values 
of  mathematics  have  always  held  an  important 
place  in  the  development  of  civilization.  The  ear- 
liest beginnings  were  with  the  utilitarian  aspect 
uppermost.  The  Egyptians  found  a  study  of  the 
elements  of  geometry  of  great  value  to  them  in 
determining  the  boundaries  of  their  lands  after  the 
Nile  inundations.  The  trading  nations,  notably 
the  Phoenicians,  needed  the  elements  of  arithmetic 
to  carry  on  successful  trading.  In  the  wars,  ap- 
plication was  constantly  made  in  war  engines,  as 
for  example  in  the  siege  of  Syracuse.  In  China 
attention  was  given  mainly  to  the  acquiring  of  a 
knowledge  of  the  simple  facts  needed  in  daily  life. 
Among  those  nations  stressing  religious  life,  only 
the  practical  values  were  studied.  Even  to  the 
peoples  of  Greece  and  Rome,  except  in  the  philo- 
sophic schools,  the  usable  facts  of  mathematics  were 
the  only  ones  of  great  value.  Not  only  were  the 
beginnings  of  mathematics  strongly  utilitarian, 
but  throughout  the  growth  of  civilization  the  prac- 
tical aspect  has  been  strongly  uppermost  in  the 
minds  of  men. 

Most  people  in  the  world  of  organized  society 
have  little  need  to  do  more  than  work  out  simple 
relationships.  But  in  all  the  myriad  affairs  of  life 
the  phenomena  of  number  have  some  indirect  bear- 
ing. So  a  knowledge  of  facts  has  its  place,  but  no 
mere  knowledge  of  facts  can  ever  show  us  the  goal 


FACTS  ONLY  MEANS  25 

of  education.  Facts  are  only  means.  The  old 
system  knew  the  goal  but  not  the  means  to  reach 
it.  Our  system  is  rich  in  means  but  often  misses 
the  goal.  Our  teachers  lose  sight  of  the  ideal  in  the 
study  of  facts  and  methods.  The  aim  of  present 
day  education  is  to  effect  such  a  compromise  as 
will  maintain  the  lofty  aims  of  the  old-time  school- 
master and  overcome  his  trials  by  the  use  of  care- 
fully devised  methods  of  teaching.  The  first  school- 
masters with  their  few  students  could  give  the  per- 
sonal touch  and  get  results  without  elaborate  method, 
but  with  the  growth  of  schools,  lack  of  time  made 
it  necessary  merely  to  hear  recitations,  and  so  re- 
sults became  less  satisfactory.  To  meet  this  sit- 
uation, the  graded  school  was  evolved,  but  here 
again  numbers  forced  the  teacher  merely  to  hear 
recitations  and  to  center  all  interest  upon  himself 
and  his  method.  Today  we  are  back  to  the  in- 
formality of  the  beginning,  attempting  to  increase 
the  efficiency  of  the  method  by  the  application  of 
modern  pedagogy.  We  hope  for  success  and  there 
is  reason  to  feel  that  we  shall  attain  it.  We  have 
the  method  and  the  ideal.  Can  we  get  the  two 
into  our  system  by  introducing  the  personal  touch 
between  teacher  and  student? 

(1)  The  facts  to  be  grouped  under  the  first  class 
are  those  of  elementary  arithmetic, — addition,  sub- 
traction, multiplication,  simple  division,  simple 
interest,  proportion  and  mensuration;  of  elemen- 
atry  geometry, — simple  construction,  method  of 
proof,  and  formulas  for  areas  and  volumes;  and, 


26  MATHEMATICS  AND  EDUCATION 

although  not  commonly  taught  in  the  grades  below 
the  high  school,  the  elements  of  algebra, — the  use 
of  the  equation,  and  letter  symbolism;  the  elements 
of  trigonometry, — computation  by  the  use  of  angu- 
lar functions.  These  facts  are  of  utmost  practical 
value  to  everyone.  Provision  should  be  made  to 
present  this  work  as  early  as  the  mental  strength 
of  students  will  warrant.  Inasmuch  as  students 
leave  school  rapidly  from  sixth  grade,  I  should  teach 
a  student  as  much  as  possible  by  the  end  of  that 
period,  even  at  the  sacrifice  of  some  of  the  mental 
training  material  if  necessary.  The  first  task  of 
the  teacher  is  always  to  produce  the  highest  quality 
of  citizenship  in  the  allotted  time.  The  production 
of  linguists  or  mathematicians  is  a  secondary  con- 
sideration. By  the  close  of  the  sixth  year  in  school, 
the  student  should  have  in  his  possession  all  the 
simple,  daily  usable  facts  of  mathematics.  He 
should  also  have  acquired  the  elements  of  spelling, 
geography,  grammar,  United  States  history,  civics, 
and  the  ability  to  read  well  and  to  write  a  good 
hand.  All  in  addition  to  this  should  be  rated  second 
from  a  standpoint  of  value.  Personal  tastes  of 
the  teacher  are  not  to  bias  the  method  and  mode 
of  teaching;  in  fact  one's  taste  should  be  to  raise 
the  standard  of  mass  efficiency  to  the  highest  during 
the  first  six  years.  Time  and  time  over,  the  ideal 
swings  back  to  the  one  expressed  by  the  dictum, 
reading,  writing,  and  arithmetic,  simply  because 
the  public  has  a  deep  seated  feeling  that  the  largest 
number  of  people  in  any  community  has  greater 


FUNDAMENTALS  FIRST  27 

need  for  the  elements  than  for  cultural  development. 
The  following  newspaper  item  may  be  read  with 
interest  concerning  this  point: 

Cut  Out  School  Frills 
The  Three  R's  Not  to  be  Lost  Sight  of  in  Chicago    \ 

Chicago,  111.,  Dec.  20. — Sweeping  reform  of  the  curriculum 
of  the  grammar  grades  of  the  city  schools  was  advocated 
yesterday  at  the  school  management  committee's  meeting. 

If  the  board  adopts  a  report  which  is  to  be  presented,  it 
will  mean  complete  elimination  from  the  schools  of  every- 
thing that  tends  to  interfere  with  the  fundamental  principles 
of  education. 

Sewing  classes,  reed  and  rattan  work  and  similar  courses, 
which  now  occupy  the  children's  hands  and  minds,  will  be 
done  away  with  if  a  motion  made  by  Trustee  John  Guerin  is 
carried. 

Mrs.  Ella  Flagg  Young,  superintendent  of  schools,  was 
openly  displeased  at  the  action  of  the  committee  in  appoint- 
ing a  subcommittee  to  report  on  the  matter.  She  made  no 
secret  of  her  feelings,  and  freely  criticised  some  of  the  com- 
mittee members. 

"I  have  been  working  toward  this  end  myself  throughout 
my  administration,"  she  exclaimed.  "Every  time  I  start 
to  do  anything  a  committee  is  appointed  to  take  the  work 
out  of  my  hands.  It  makes  one  wonder  if  a  superintendent 
can  accomplish  anything." 

The  motion  made  by  Dr.  Guerin  that  "everything  that 
tends  to  interfere  with  the  teaching  of  reading,  writing, 
arithmetic,  spelling,  geography,  physiology,  grammar  and 
United  States  history  be  eliminated  from  the  school  system" 
was  seconded  by  Julius  Smeitanka.  Every  member  of  the 
committee  expressed  himself  as  being  unqualifiedly  in  favoJ 
of  taking  "mollycoddle  trivialities"  from  the  schools. 

From  the  attitude  of  the  members  there  was  apparently 
no  question  that  the  elimination  would  take  place.  Even 
those  appointed  on  the  sub-committee,  while  asserting  that 
they  were  barred  from  speaking  for  publication  because 
they  were  members  of  a  committee,  did  not  hesitate  to  de- 
clare themselves  individually  for  the  reform. 

(2)  The  second  group  of  facts  has  contingent 
and  cultural  value.  It  embraces  those  facts  under- 
lying the  greater  institutions  of  our  civilization, 


MATHEMATICS  AND  EDUCATION 

as  the  railroad,  the  ocean  liner,  the  sky-scraper, 
the  bridge,  the  telephone,  the  telegraph,  the  auto- 
mobile, the  wireless,  the  airship,  automatic  machin- 
ery, the  elements  of  astronomy,  etc.  etc;  the  facts 
of  the  historical  development  of  mathematics;  the 
facts  of  the  common  trades  and  vocations  with 
mathematical  connections;  and  the  facts  of  busi- 
ness intercourse.  It  promotes  a  student's  general 
j  aptitude  for  life,  to  know  something  of  almost  any 

I  line  of  work  he  may  enter.  He  has  a  fund  of  power 
which  can  be  used  in  a  great  variety  of  ways.  The 
student  seldom  knows  exactly  what  his  life  work 
is  to  be;  in  fact  ninety  percent,  of  those  who  choose 
a  line  of  work  while  young  undergo  change  of  mind. 
It  is  advantageous  for  the  student  to  have  a  variety 
of  material  presented,  so  that  he  may  discover 
his  aptitude  for  one  particular  line.  This  group  of 
facts  also  promotes  the  culture  of  the  student  more 
than  any  other  group.  It  provides  informational 
value  upon  many  lines.  It  gives  a  more  intimate 
relation  to  nature  and  her  phenomena,  thus  open- 
ing a  wider  and  better  world  to  the  student.  It 

i  encourages  the  appreciation  of  the  triumphs  of  the 

j  human  mind.  It  gives  him  a  more  kindly  feeling 
toward  the  well-related  and  beautiful.  It  guaran- 
tees a  stock  of  thoughts  tending  to  make  points  of 
contact  between  him  and  passing  acquaintance, 

Hhus  furnishing  material  for  conversation.  In  short, 
this  group  of  facts  aids  him  to  acquire  self  perfec- 
tion along  general  lines,  to  react  to  better  advantage 
to  society  and  to  nature.  From  society's  stand- 


THE  THREE  GROUPS  OF  FACTS  29 

point  there  is  increased  opportunity  opened  to  each 
individual.  The  facts  falling  under  this  heading 
are  all  those  general  facts  ccvering  the  manual  arts, 
the  natural  sciences,  the  growth  of  the  science  of 
mathematics,  the  business  world,  and  the  mechani- 
cal arts.  They  are  those  facts  which  enter  into  all 
thorough  discussion  of  the  science,  and  which  the 
student  acquires  without  special  attention  being 
given  to  their  projection.  They  are  the  facts  the 
students  are  bound  to  get,  providing  the  teacher 
carries  on  his  work  carefully. 

(3)  The  third  group  of  facts  consists  of  those 
involved  in  the  great  engineering  successes,  such 
as  electrical  engineering;  in  the  promotion  of  large 
business  transactions;  and  in  the  development  of 
the  science  for  its  own  sake.  While  it  is  true  that 
96%  of  the  students  in  school  do  not  reach  the 
university  and  need  no  special  training  to  pursue 
their  chosen  vocation,  the  remaining  4%  do  get 
into  the  universities  and,  strange  as  it  may  seem, 
this  4%  rules  the  world.  It  is  necessary  therefore 
to  keep  the  door  of  opportunity  open  to  that  par- 
of  this  group  who  will  fill  the  high  positions  of  the 
engineering,  business,  and  scholarly  world.  Since 
this  group  of  facts  is  presented  only  in  the  higher 
educational  institutions,  the  ordinary  teacher  is 
not  especially  interested,  except  is  so  far  as  prepa- 
ration is  concerned.  On  this  point  little  need  be 
said.  If  the  bright  student  is  allowed  to  advance 
and  finds  a  helpful  and  suggesting  friend  in  the 
teacher,  he  asks  nothing  more. 


LOS  ANGELES 

STATE  NORMAL  SCHOOL 


30  MATHEMATICS  AND  EDUCATION 

Mathematics  typifies  a  method  of  thought.  It 
stands  for  a  clear,  direct  development  to  a  valid 
conclusion  from  accepted  premises.  Whether  the 
method  be  synthetical  or  analytical,  inductive  or 
deductive,  the  argument  is  exact  and  sure  of  its 
end:  so  sure  in  fact,  that  no  matter  what  doubts 
one  has  as  to  the  certainty  of  other  things,  there 
is  never  a  doubt  that  two  plus  two  equals  four.  So 
wonderfully  potent  is  it  as  a  type  of  thought  that 
men  swear  by  it.  In  argument  they  put  their  faith 
in  numerical  statistics,  confident  that  their  hearers 
will  accept  as  valid  that  which  is  based  on  number 
relation.  As  direct  results  of  the  acquirement  of 
this  method  of  thought  several  values  come  to  the 
students.  They  are,  (1)  the  habit  and  ability  to 
draw  conclusions  from  given  data,  (2)  the  appre- 
ciation of  the  necessity  of  a  conclusion,  (3)  the  ap- 
preciation of  a  logical  development  back  of  every 
v  conclusion,  (4)  the  acquisition  of  an  effective  method 
in  other  scientific  subjects,  and  (5)  the  ability  to 
generalize  conceptions.  Indirectly  the  develop- 
ment of  such  a  type  of  thought  gives  power  in  (1) 
grasping  situations,  (2)  use  of  symbolic  language, 

(3)  appreciation  of  the  great  advantage  in  validity, 

(4)  habits  of  self  study,  (5)  in  development  of  men- 
.    tal  strength,  (6)  in  developing  habits  of  quickness, 

\  neatness,    orderly    arrangement,    reflection,    steadi- 
ness, and  certainty. 

These  various  values,  mentioned  under  the  group 
of  direct  results,  are  usually  passed  off  as  falling 
under  the  culture  category  and  there  the  argument 


HABIT  OF  LOGICAL  THOUGHT  31 

ends.  As  a  matter  of  pertinent  fact  these  values 
are  fully  as  practical  and  utilitarian  as  the  fact 
values.  Perhaps  more  so,  for  the  facts  can  be  taken 
direct  from  reference  books  if  the  demand  is  suf- 
ficient to  make  it  necessary  to  have  them,  while 
the  habit  of  logical  thought  can  only  be  developed. 
One  of  the  greatest  assets  a  person  can  have  in  any 
line  of  work  is  ability  to  draw  quick  and  certain 
conclusions  from  given  data.  In  every  occupation 
situations  constantly  form  themselves  to  be  solved 
by  the  man  or  woman  who  has  the  faculty  of  quick 
thinking  and  quick  acting.  The  world  seems  to 
have  endless  opportunities  for  just  this  sort  of  per- 
son, and  although  the  slow  ones  invariably  attribute 
the  advancement  of  their  co-workers  to  luck,  the 
real  reason  is  in  their  superior  ability  to  arrange 
the  data  into  details  and  fundamental  principles 
and  then  draw  a  quick  conclusion.  Nearly  all  large 
business  transactions  are  made  upon  newly  created 
conditions,  with  the  advantage  always  with  the  one 
who  thinks  the  most  rapidly.  To  be  able  to  see 
the  conclusions  involved  in  changing  conditions  is 
in  itself  power. 

Not  alone  is  there  power  in  ability  to  draw  a 
conclusion,  but  likewise  in  the  appreciation  that  a 
conclusion  is  necessary.  Whatever  the  conditions 
of  a  situation  are  or  whatever  the  particular  phase 
of  social  activity  the  data  has  reference  to,  there 
must  be  a  necessary  conclusion.  It  is  not  always 
appreciated  by  the  ordinary  thinker,  that  a  given 
set  of  data  necessitates  a  conclusion.  Thus  it  is 


32  MATHEMATICS  AND  EDUCATION 

that  so  many  men  and  women,  though  excellent 
workers,  fail  to  succeed  when  given  responsibility. 
They  have  power  to  handle  extensive  detail  but  no 
power  of  organization,  owing  to  lack  of  ability  to 
grasp  the  significance  of  similar  characteristics. 
In  our  particular  civilization  there  is  a  strong  feeling 
toward  compilation  of  statistics.  This  is  a  direct 
example  of  the  appreciation  of  the  necessity  of  a 
conclusion.  Statistics  are  compiled  to  make  pos- 
sible the  drawing  of  a  conclusion.  Were  it  not  for 
the  fact  that  a  feeling  exists  that  a  conclusion  is 
implicitly  expressed,  there  would  be  no  value  in 
statistics.  Young  men  in  business  life  who  have 
confidence  in  the  necessity  of  a  conclusion  are  con- 
stantly on  the  alert  to  discover  one.  They  study 
the  detail  of  their  department  and,  it  is  needless  to 
say,  they  succeed.  For  one  to  be  trained  to  feel 
the  necessity  of  a  conclusion  is  to  make  for  efficiency. 
It  is  a  practical  value  and  it  is  a  cultural  value  as 
well.  Then  too,  the  conclusion  may  be  given.  It 
is  power  to  know  that  back  of  it  is  a  logical  develop- 
ment. It  is  the  first  step  toward  analysis.  In  the 
activities  of  everyday  life  one  is  continually  asking 
why  the  particular  conclusion  is  valid.  The  only 
test  is  to  consider  the  development  leading  to  the 
conclusion  and  test  each  step  for  validity.  To  know 
that  the  conclusion  has  a  logical  development  is 
evidently  essential. 

Mathematics  is  most  valuable  in  giving  students 
their  first  insight  into  the  method  of  inference  and 
approximation.  With  no  claim  that  the  training 


GENERALIZING   POWER  33 

can  be  carried  over  directly  (which  offends  many 
educators),  it  is  safe  to  say  that  one  finds  in  mathe- 
matics a  reasoning  to  conclusions  which  is  so  simple 
and  certain  that  it  may  be  used  with  profit  as  a 
beginning  for  such  training.  Every  event  or  fact, 
regardless  of  the  outward  appearance  of  contin- 
gency, has  a  development  which  one  trained  in 
mathematics  readily  understands.  There  are  an- 
tecedent events  and  .facts,  perhaps  not  evident, 
but  nevertheless  existent,  which  lead  to  the  conse- 
quent. It  is  surprising  to  note  the  simplicity  of  an 
analysis  once  there  is  method  and  appreciation  of 
necessary  antedecents.  Mathematics,  owing  to  its 
certainty,  gives  perhaps  the  best  elementary  train- 
ing in  such  mental  activity.  In  other  sciences  a 
method  is  absolutely  necessary  to  develop  working 
laws;  since  work  in  them  rarely  starts  before  the 
eighth  or  ninth  grades,  the  method  may  be  develop- 
ed through  the  study  of  mathematics. 
/  One  of  the  characteristics  of  a  strong  mind  is  the 
/generalizing  power,  organization  of  details  and  ar- 
rangement under  heads  and  sub-heads.  The  great- 
est movement  in  the  world  today  is  toward  organiza- 
tion. It  is  in  the  air.  Trusts  organize  under  the 
leadership  of  men  of  genius  until  they  control  the 
whole  of  the  social  group.  In  every  occupation 
the  tendency  is  to  organize.  The  results  of  such 
organization  may  be  good  or  bad,  but  if  education 
is  to  follow  the  conscious  aim  of  society  there  must 
be  that  in  the  materials  of  education  and  the  pro- 
cesses of  education  which  tends  to  •  develop  such 


34  MATHEMATICS  AND  EDUCATION 

power.  Is  it  too  much  to  claim  that  the  study  of 
mathematics  is  one  of  the  best  means  to  such  an 
end, — that  is  when  mathematics  is  taught  as  a  liv- 
ing science  rather  than  a  dead  one,  and  when  stu- 
dents are  taught  to  evolve  their  own  mathematics? 
Unfortunately  there  are  those  among  us  who 
fail  to  distinguish  between  formalism  and  business 
method.  Fortunately  they  are  in  the  minority. 
Otherwise  the  work  of  education  could  not  have 
reached  its  present  state  of  efficiency.  In  educa- 
tion, as  in  business,  detail  must  hold  a  subordinate 
place  and  systematic  handling  of  this  detail  makes 
for  economy  in  time  and  energy.  Modern  busi- 
ness has  appreciated  the  importance  of  system  to 
the  extent  of  constructing  an  office  that  approxi- 
mates a  machine  in  its  exactness  and  quickness. 
If  he  whose  raw  material  is  iron  and  wood  finds  it 
advantageous  to  give  its  greater  attention  to  funda- 
mentals, how  much  more  advantageous  is  it  to  him 
whose  raw  material  is  the  growing  mind  of  the  com- 
munity, to  put  his  greater  attention  on  the  funda- 
mentals?' A  business  man  not  only  decreases  ex- 
5 Dense  and  time  by  systematic  methods,  but  he  im- 
Droves  the  quality  of  business  done.  So  it  is  with 
the  teacher.  It  has  been  said  with  truth,  that  a 
student  on  one  end  of  a  log  and  a  teacher  upon  the 
other  constitutes  a  school.  Unfortunately,  how- 
ever, modern  school  systems  are  not  so  ideal.  There 
are  forty  students  for  one  teacher  in  the  modern 
school.  Thus  system  in  handling  is  made  essential. 
Not  alone  in  the  administrative  department  is  econ- 


POWER  TO  GRASP  SITUATIONS  35 

omy  of  time  and  energy  desirous  but  in  every  school- 
room as  well.  Each  class  must  have  its  individual 
unity  as  well  as  its  subordinate  place  in  the  larger 
system.  It  should  have  its  individual  organization. 

The  study  of  any  group  of  facts  touching  daily 
life,  whether  classified  as  history,  language,  mathe- 
matics, or  what  not,  is  educative.  There  are  de- 
velopments which  the  student  gets  regardless  of 
the  particular  content.  Among  these  may  be  men- 
tioned the  power  to  grasp  situations,  the  power  of 
yusing  symbolic  language,  the  power  of  testing  for 
\validity,  the  power  of  self-study,  and  the  power 
/developing  habits  of  quickness,  neatness,  orderly 
|  arrangement,  reflection,  steadiness,  and  certainty. 
It  is  not  claimed  that  only  through  a  study  of  mathe- 
matics can  these  developments  be  attained,  but  it 
is  claimed  that  the  study  of  mathematics  is  one  of 
the  most  effective  means  to  such  ends.  Extensive 
discussion  upon  this  point  is  not  necessary.  These 
developments  are  admitted  by  all  thinking  people, 
just  as  they  are  admitted  by  mathematicians  for 
all  other  subjects.  They  are  values  the  student 
gets  from  systematic  study  along  any  line.]  <  .4 

To  study  mathematics  for  its  own  sake  is  equiva- 
lent to  studying  mathematics  for  the  various  values 
mentioned,  plus  mental  pleasure.  We  study  art, 
music,  literature,  language,  political  economy,  in 
fact  all  subjects,  partly  for  their  value  in  applica- 
tion and  partly  because  of  the  pleasure  they  afford. 
The  pleasure  value  is  always  a  large  one  unless  we 
desire  to  make  of  ourselves  mere  machines.  Man 


36  MATHEMATICS  AND  EDUCATION 

does  not  live  merely  to  eat  and  sleep.  His  educa- 
tion should  not  be  merely  the  means  of  acquiring 
food,  clothing,  and  shelter.  Essential  as  these 
things  are,  a  human  life  has  a  much  deeper  meaning 
than  this.  The  full  measure  of  human  happiness 
comes  with  a  love  for  the  beautiful  and  ideal  in  all 
things.  That  education  then  is  not  complete, 
which  does  not  touch  upon  the  ideal.  In  mathe- 
matics, as  in  all  else,  there  is  a  beauty  and  an  ideal 
condition.  It  does  give  mental  pleasure  to  have 
the  power  to  appreciate  this.  Is  it  then  in  vain, 
that  the  love  of  a  perfect  demonstration  is  encour- 
aged? Not  unless  we  limit  our  educational  ideal 
to  cover  the  crudities  of  life.  To  live  is  not  a  diffi- 
cult thing  in  this  world;  to  live  joyfully  is  always 
the  difficult  thing  to  do.  There  is  a  value  in  study- 
ing mathematics  for  its  own  sake  just  as  there  is  a 
value  in  studying  any  other  subject  for  its  own  sake. 
The  only  point  to  remember  is  that  the  utilitarian 
value  must  be  first. 

If  education  for  the  group  means  such  organization 
as  will  give  greatest  opportunity  for  the  individual 
to  realize  his  ends,  then  the  individual,  as  a  factor 
of  the  group,  must  become  aware  of  the  necessity 
of  the  individual's  appreciation  of  what  is  meant 
by  the  word  welfare.  There  must  be  in  the  educa- 
tion of  the  individual,  that  training  which  promotes 
his  desire  to  think  and  do  what  is  best  for  the  group 
life.  He  needs  to  feel  that  that  action  which  alone 
seems  to  promote  his  own  well  being,  is  not  entirely 
what  it  seems;  for  indirectly  be  becomes  a  hindrance 


INDIVIDUAL  THINKING  37 

to  better  organized  life,  not  only  through  his  own 
negative  action  but  through  the  example  which 
prompts  others  to  do  likewise.  The  social-welfare 
side  of  education  must  combat  not  only  the  ten- 
dencies of  individuals  toward  selfish  action,  but 
also  age-long  customs  inaugurated  by  primeval 
group  life.  This  phase  of  education,  though  well 
enough  understood,  is  extremely  difficult  for  the 
teacher  to  stress.  It  is  not  something  which  can  be 
taught  directly  but  rather  must  be  presented  in  an 
incidental  manner  through  constant  suggestion  and 
example.  It  is  difficult  to  maintain  an  approxi- 
mation to  the  ideal  among  adults  who  have  the 
responsibilities  of  citizenship  upon  them.  How  much 
more  difficult  for  care-free  children  to  find  an  in- 
terest in  things  of  this  sort ! 

In  fact,  there  is  more  or  less  loose  thinking  con- 
cerning this  point  on  the  part  of  teachers.  The 
fact  that  teachers  are  underpaid  and  overworked 
does  not  help  them  to  have  full  conviction  of  their 
professed  faith.  They  find  it  is  often  almost  neces- 
sary to  sacrifice  the  ideal  for  individual  betterment. 
It  is  merely  proof  of  the  general  statement  that  in- 
dividual welfare  is  most  prominent  in  the  individual 
mind. 

In  this  connection,  the  encouragement  of  indi- 
vidual thinking  on  the  part  of  the  student  is  of  great 
value.  Every  person  is  endowed  with  different 
abilities.  To  get  the  best  reaction,  individuality 
is  essential.  Only  through  the  group  life  can  man 
find  his  highest  development,  and  the  highest  par- 


38  MATHEMATICS  AND  EDUCATION 

ticipation  in  group  life  is  possible  only  for  him  who 
emphasizes  his  own  uniqueness.  The  modern  ethi- 
cal conception  is,  that  ethical  laws  exist  for  man. 
The  center  of  attention  is  not  the  form  but  the 
substance.  The  student  must  realize  the  power 
of  originality  in  thought  and  action.  In  a  system 
of  free  education,  to  which  the  entire  community 
contributes,  training  in  citizenship  is  not  only  ex- 
pected but  demanded.  Those  qualities  making 
for  a  better  and  closer  relationship  between  the  in- 
dividual and  the  community,  stand  out  prominent- 
ly as  qualities  to  be  desired  in  the  student.  It  is 
an  expression  of  the  very  essence  of  democracy. 
1  Education  must  have  as  a  part  of  its  aim  the 
common  good.  Modern  society  is  in  a  stage  of 
conscious  development.  So  well  understood  is 
this  that  education  is  denned  as  a  process  of  adjust- 
ment. Mere  knowledge  in  its  narrow  sense  is,  then, 
not  sufficient.  There  must  be  training,  covering 
points  of  social  activity  and  also  of  leisure,  for  social 
welfare  is  as  much  concerned  with  individuals  out 
of  action  as  with  those  in  action. 

But  the  greatest  value,  perhaps,  is  derived  from 
the  definition  of  mathematics  as  a  quantitative  ap- 
preciation of  the  world  in  which  we  live.  Denned 
in  this  way  mathematics  becomes  a  point  of  view, 
and  has  its  application  in  all  things.  Not  only  is 
it  possible  to  find  special  application  in  the  various 
problems  of  science,  history,  language,  etc.,  but 
also  to  study  these  branches  of  knowledge  mathe- 
matically. It  is  evident  that  such  a  definition  holds 


MATHEMATICS  AS  A  POINT  OF  VIEW          39 

true  only  when  one  supposes  that  the  world  in  all 
its  aspects  can  be  thought  of  quantitatively.  Such 
supposition  is  probably  universally  held.  Also  it 
supposes  that  all  situations  must  be  judged  from 
one  point  of  view  at  a  time,  the  quantitative  point 
of  view  being  one.  Just  as  things  are  considered 
from  the  artistic  point  of  view,  the  utilitarian  point 
of  view,  or  the  ethical  point,  so  can  they  be  con- 
sidered from  a  quantitative  point  of  view. 

If  mathematics,  arithmetic  in  particular,  is  de- 
nned in  this  way,  its  great  value  includes  the  various 
ones  mentioned,  all  of  which  function  directly  to- 
ward the  perfection  of  the  general  value.  All  sub- 
jects may  be  considered  as  points  of  view.  His- 
tory, for  instance,  judges  the  events  with  reference 
to  events  which  have  taken  place,  events  which  are 
contemporary,  and  events  which  may  take  place 
in  the  future, — it  judges  an  event  with  regard  to 
its  place  in  a  sequence.  Mathematics  may  in  this 
way  become  vitally  connected  with  all  school  ac- 
tivity. This  value  need  not  detract  from  the  values 
of  the  subject  as  a  distinct  science. 

With  the  emphasis  placed  upon  the  idea  of  mathe- 
matics as  a  point  of  view  there  is  given  a  basis  for 
reconstruction.  Only  those  processes  having  most 
common  application  should  be  stressed,  and  all 
processes  having  the  character  of  mental  gymnas- 
tics should  be  eliminated.  The  following  list  of 
omissions  are  suggested  by  McMurry,  "apothe- 
caries' weight,  troy  weight;  examples  in  longitude 
and  time  except  the  very  simplest,  involving  the 


40  MATHEMATICS  AND  EDUCATION 

15°  unit,  since  our  standard  time  makes  others  un- 
necessary; the  furlong  in  linear  measure;  the  rood 
in  square  measure;  the  dram  and  the  quarter  in 
avoirdupois  weight;  the  surveyor's  table;  table  on 
folding  paper;  all  problems  in  reduction,  ascending 
and  descending,  involving  more  than  two  steps; 
the  G.  C.  D.  as  a  separate  topic,  but  not  practice  in 
detecting  divisibility  by  2,  3,  5,  and  10;  all  work 
with  L.  C.  M.,  except  of  such  very  common  denomi- 
nators as  those  just  mentioned;  complex  and  com- 
pound fractions  as  separate  topics;  compound  pro- 
portion; percentage  as  a  separate  topic,  with  its 
cases;  true  discount;  most  problems  in  compound 
interest,  and  all  in  annual  interest;  problems  in 
partial  payments,  except  those  of  a  very  simple 
kind;  the  same  for  commission  and  brokerage,  for 
example,  all  problems  involving  fractions  of  shares; 
profit  and  loss  as  a  special  topic;  equation  of  pay- 
ments— made  unnecessary  by  improved  banking 
facilities;  partnership — made  unnecessary  in  the 
old  sense  by  stock  companies;  cube  root;  all  algebra, 
except  such  simple  use  of  the  equation  as  is  directly 
helpful  in  arithmetic ;  in  addition  to  all  this,  arithme- 
tic may  be  omitted  as  a  separate  study  throughout 
the  first  year  of  school,  on  the  ground  that  there 
is  no  need  of  it,  if  the  number  incidentally  called 
for  in  other  work  is  properly  attended  to." 

With  such  omissions  made  there  would  be  ample 
time  for  full  consideration  of  all  subjects  quantita- 
tively, that  is  mathematically.  Even  with  a  list 
of  omissions  less  complete,  more  time  could  be  de- 


MATERIALS  OF  MATHEMATICS  41 

voted  to  the  subjects  which  directly  concern  us  in 
daily  life.  When  one  considers  that  education  aims 
not  so  much  toward  the  acquisition  of  a  body  of 
facts  as  toward  power  to  interpret  these  facts  in 
the  light  of  serviceableness,  mathematics  becomes 
a  functioning  process,  as  do  all  other  subjects. 

Materials  of  Mathematics 

From  the  preceding  discussion  the  power  for 
successful  reaction  of  the  individual  calls  for  (1) 
a  group  of  facts  so  basic  as  to  be  easily  applied  to 
all  human'activity  and  so  well  mastered  as  to  be 
available  for  all  situations,  (2)  a  group  of  facts 
which,  although  not  used  in  every  application,  may 
be  used  in  part  of  every  application  (contingent 
facts),  (3)  a  group  of  facts  used  only  in  advanced 
scientific  work,  as  for  example,  engineering,  (4)  a 
method  of  clear,  and  simple  reasoning  leading  from 
given  premises  to  a  sure  conclusion,  (5)  an  appre- 
ciation of  the  necessity  of  a  conclusion  where  a  se- 
quence of  events  is  given,  (6)  ability  to  analyze  a 
situation  and  see  the  sequence  relation  between 
parts,  (7)  generalization  and  organization  of  de- 
tails, and  (8)  habits  conducive  to  clear  thinking, 
such  as  those  of  quickness,  exactness,  neatness, 
steadiness,  etc. 

The  materials  of  mathematics  must  be  selected 
carefully,  not  only  to  obtain  that  which  is 
best  suited  to  production  of  power  for  reaction,  but 
also  to  give  during  certain  time  limits  that  material 
best  suited  to  the  varying  strength  of  the  pupil. 
Also,  the  material  must  be  selected  with  respect 


42  MATHEMATICS  AND  EDUCATION 

for  the  fact  that  students  start  dropping  out  of 
school  rapidly  by  the  end  of  the  sixth  year.  Unless 
the  teacher  finds  an  interest  in  these  points  there 
is  educational  waste  and  general  inefficiency.  The 
materials  of  mathematical  education  are  (1)  those 
of  arithmetic,  and  (2)  those  of  the  advanced  courses, 
including  algebra,  geometry,  trigonometry,  analytic 
geometry,  and  calculus.  Because  of  the  fact  that 
all  students  study  arithmetic,  while  only  a  rela- 
tively small  number  study  the  advanced  courses, 
and  also  because  of  the  fact  that  arithmetic  is  stud- 
ied from  six  to  eight  times  as  long  as  any  one  of 
the  other  courses,  it  is  evidently  our  duty  to  devote 
the  greater  part  of  our  discussion  to  it.  With 
better  citizenship  given  as  the  aim  of  education, 
arithmetic  must  stand  or  fall  upon  its  merits  with 
reference  to  this  thing.  Age  long  customs  cannot 
be  defended  upon  mere  sentimental  grounds.  Un- 
less arithmetic  contributes  proportionately  to  the 
time  and  energy  expended  upon  it,  there  must  be 
change  either  (1)  in  methods  of  teaching,  (2)  in 
materials  used,  or  (3)  in  both. 

Education  toward  efficient  citizenship  must  be 
through  purposeful  work.  All  other  work  becomes 
drudgery  and  drudgery  is  deadening.  It  is  true, 
no  doubt,  that  we  get  our  number  ideas  from  meas- 
urement, comparison,  and  relationship  of  things. 
The  number  idea  is  generated  within  the  mind  by 
constructive  activity.  This  theory  opposes  the 
theory  that  number  is  something  perceived,  and 
makes  number  a  creation  of  the  mind. 


ARITHMETIC  43 

Measurement  of  things  is  the  basic  operation  or 
activity  and  the  first  to  be  performed  in  order  of 
time.  Out  of  the  physical  act  of  placing  the  hand 
upon  objects  and  at  the  same  time  speaking  words, 
(number  names),  or  pointing  to  objects  and  speak- 
ing words,  (number  names),  — the  first  activity  of 
•  a  child, — the  unit  idea  must  develop.  Then  com- 
parison of  the  unit  idea  with  the  whole  idea,  and 
finally  the  ideas  of  analysis,  synthesis,  and  relation. 
Out  of  the  mechanical  activity  of  counting  grows 
the  idea  of  measurement,  or  application  of  the  unit 
idea.  Measurement  is  impossible  without  the  unit, 
so  in  some  manner  the  beginning  idea  of  the  unit 
is  developed  through  mechanical  counting.  Just 
how,  it  is  probably  impossible  to  say.  There  is  a 
development  from  the  counting  of  mere  separate 
entities,  to  the  counting  of  separate  entities  having 
common  characteristics.  The  next  step  is  one  of 
analysis  of  given  wholes  into  evident  entities.  This 
is  no  doubt  a  mechanical  operation  in  most  part. 
Synthesis  and  organization  is  the  next  step,  and 
finally  comes  relation.  During  the  counting  pro- 
cess the  number  names  are  learned.  The  second 
step,  analysis,  brings  out  the  idea  that  the  names 
refer  to  parts  of  larger  things.  It  is  through  the 
process  of  counting  objects  with  common  character- 
istics and  through  this  elementary  analysis,  that 
the  idea  of  a  unit,  a  standard  for  measurement,  is 
first  generated.  The  fundamental  operations  bring 
out  synthesis  and  organization.  The  work  involv- 
ing the  relative  size  of  quantities  with  regard  to 


44  MATHEMATICS  AND  EDUCATION 

the  same  unit,  relative  size  of  quantities  with  regard 
to  different  units,  change  of  unit,  etc.,  bring  out 
the  idea  of  relation. 

Following  this  outline  the  whole  development 
of  the  number  idea  is  worked  out.  The  materials 
of  this  development  must  be  selected  with  regard 
to  these  steps.  It  is  quite  impossible  to  give  exact 
limits  for  the  development  for  each  grade,  owing 
to  differences  in  children  and  the  differences  in 
teachers,  but  in  general  the  work  may  be  blocked 
out  as  follows: 
Kindergarten 

All  work  should  be  incidental  with  counting  and 
grouping,  brought  in  as  occasion  arises.  This  is 
to  depend  entirely  upon  the  judgment  of  the  teacher 
and  the  age  of  children.  But  by  all  means  the 
counting  started  at  home  should  be  continued,  and 
continued  toward  unit  appreciation.  The  word 
"half"  may  be  used,  and  perhaps  "quarter"  under 
proper  conditions.  Paper  folding,  beads,  blocks, 
etc.  give  material  for  such  work. 
First  grade 

Counting  should  be  continued  and  application 
made  incidentally  to  all  possible  things,  as  for  in- 
stance, the  number  in  the  class,  the  pieces  of  mater- 
ial used, — as  paper,  pencils,  rulers,  clock  numerals, 
days  in  week,  and  number  as  it  occurs  in  other 
subjects,  etc.,  etc.  Very  little  should  be  done  to- 
ward developing  the  number  work  as  a  thing  apart. 
Here  again  the  judgment  of  the  teacher  must  be 
the  deciding  factor.  The  advance  made  upon  the 


ARITHMETIC  BY  GRADES  45 

kindergarten  should  be  to  remove  counting  from 
the  merely  mechanical  operation  to  a  purposive 
operation.  The  work  in  grouping  and  in  combina- 
tions should  be  carried  on,  with  the  idea  of  making 
the  mathematics  a  quantitative  point  of  view. 
Second  grade 

The  incidental  work  should  be  continued  and  the 
number  space,  from  one  to  fifteen  or  twenty,  dis- 
cussed according  to  strength  of  students  and  the 
ability  of  the  teacher.  In  this  work  the  counting 
may  be  made  to  embrace  the  fives  and  tens.  As 
formerly,  the  application  should  be  to  common 
objects. 

In  number  conbinations  the  addition  and  sub- 
traction series  should  precede  the  multiplication 
series.  With  the  multiplication  series  the  begin- 
ning of  the  multiplication  table  may  be  started. 

In  the  application  the  use  of  standard  units  should 
be  encouraged.  Also  there  should  be  the  one  to 
one  correspondence  between  name,  number-picture, 
and  the  figures  or  symbol. 

The  blocks,  splints,  clock  face,  (a  study  of  the 
Roman  symbols  from  one  to  twelve  should  be  made 
to  give  the  student  the  ability  to  read  spaces),  and 
units  in  linear,  dry,  and  liquid  measure  are  presented 
through  the  use  of  apparatus;  all  are  useful  not  for 
their  own  sake  but  to  help  present  the  desired  idea. 

Throughout  the  first  two  grades  the  oral  work 
should  predominate.  It  is  not  advisable  that  all 
written  or  seat  work  should  be  left  to  the  later 
grades  because  education  is  attained  through  the 


46  MATHEMATICS  AND  EDUCATION 

use  of  all  the  senses,  but  there  can  be  no  doubt  of 
the  fact  that  very  much  of  this  is  impractical.  Sup't 
Wright  of  Michigan  goes  so  far  as  to  say,  "The  en- 
tire time  of  each  class  period  throughout  the  first 
four  grades  should  be  devoted  to  oral  work,  and  no 
seat  work  should  be  given.  Written  work  may  be 
introduced  in  the  fifth  grade  and  continued  on 
throughout  the  course.  However,  the  greater  part 
of  the  work  up  to  the  seventh  grade  should  be  oral. 
Give  a  daily  oral  drill  in  every  grade.  Seat  work, 
unless  carefully  watched  and  a  time  limit  put  on 
the  class,  leads  to  inaccurate,  slow,  and  untidy 
habits  of  work,  and,  except  in  advanced  work,  in- 
creases neither  the  mental  power  nor  arithmetical 
skill." 

To  summarize,  the  work  of  the  first  two  grades 
should  cover  counting,  reading,  and  writing,  within 
the  number  space  from  one  to  one  hundred. 

The  number  space  from  one  to  twenty  may  be 
studied  through  addition  and  subtraction  tables. 

The  multiplication  table  may  be  studied  through 
the  fives  up  to  ten  and  in  cases  to  twelve. 

Division  may  be  introduced  as  the  inverse  opera- 
tion of  multiplication. 

Fractional  terms  may  be  used,  such  as  £,  $,  i, 
and  perhaps  -5-.  Students  are  usually  familiar  with 
the  terms  from  home  training,  so  it  is  desirable  to 
keep  this  connection  with  the  home  work. 

The  units  of  linear,  liquid,  and  dry  measure  may 
be  introduced  in  incidental  ways,  and  if  occasion 
arises  the  signs  +,  — ,  X,  and  H-. 


ARITHMETIC  BY  GRADES  47 

It  is  a  debatable  point,  whether  the  work  should 
go  beyond  this  point,  and  much  depends  upon  the 
strength  of  the  children  and  upon  the  skill  of  the 
teacher. 

In  some  cases  addition  may  be  carried  on  to 
cover  two  and  three  place  numbers  with  simple 
carrying,  with  subtraction  as  an  inverse  process. 
Also  division  may  be  introduced,  provided  there  is 
no  remainder.  Counting  may  be  carried  on  as  far 
as  students  are  able. 
Third  grade 

All  the  work  of  the  second  grade  should  be  re- 
viewed, and  series  formed  for  addition  and  subtrac- 
tion in  the  space  from  one  to  twenty. 

The  counting  should  be  reviewed  to  one  hundred 
and  also  by  2's,  4's,  5's,  10's,  etc.  This  is  aimed 
to  make  the  student  familiar  with  the  number  names 
and  symbols.  Following  the  mere  counting  must 
come  addition  and  subtraction  series  and  an  advance 
upon  the  multiplication  table.  Three  to  six  place 
numbers  may  be  added  and  subtracted. 

Drill  work  in  the  early  grades  is  important,  not 
for  the  mental  discipline  so  much,  as  for  the  famil- 
iarity with  elementary  facts,  needed  to  carry  on 
satisfactory  work  within  time  limits. 

The  decimal  scale  may  be  introduced  and  coins 
used  for  illustrative  material.  In  this  connection 
the  reading  and  writing  of  large  numbers,  together 
with  pointing  off,  is  profitable. 

Compound  numbers  also  may  be  introduced  with 
measurements  in  pounds  and  ounces;  quart  and 


48  MATHEMATICS  AND  EDUCATION 

peck;  minute,  hour,  and  day;  square  measure,  etc. 

Throughout  the  year  every  possible  application 
should  be  made  to  home  duties,  involving  building, 
gardening,  buying  and  selling,  etc.;  to  other  sub- 
jects of  study,  such  as  simple  geography,  natural 
science,  story  telling,  etc.;  to  the  childhood  games, 
including  score  keeping,  etc. 

During  the  work  in  the  first  three  grades  sense 
training  is  of  greatest  value.  Courses  of  study  are 
always  at  hand  for  the  teacher,  and  in  general  should 
be  followed.  The  teacher,  however,  is  the  ruling 
factor  always.  Her  judgment  is  of  greatest  value, 
for  every  school  room  is  filled  with  a  different  group 
of  students,  and  to  her  is  left  the  decision  as  to  how 
the  material  should  be  presented.  In  communi- 
ties where  the  foreign  population  is  large  the  home 
preparation  is  often  inferior.  Here  the  teacher 
finds  the  course  of  study  to  to  be  followed  accord- 
ing to  the  strength  of  the  pupils. 
Summary : 

Addition  through  six  to  eight  place  numbers, 
together  with  drill  on  combinations. 

Subtraction  of  numbers  involving  simple  borrow- 
ing. 

Multiplication  table  to  tens  or  twelves,  as  strength 
of  students  warrants.  Problems  involving  bor- 
rowing may  be  given. 

Division  following  the  multiplication  table,  and 
short  division  when  there  are  no  remainders.  •*•» 

Introductory  work  in  fractions  limiting  the  work 
to  unit  fractions. 


ARITHMETIC  BY  GRADES  49 

Denominate    numbers    involving    the    standard 
units  and  simple  measurement. 
Fourth  grade 

Not  only  review  in  addition  and  subtraction,  but 
in  multiplication  and  the  multiplication  table. 
Numbers  of  two  and  three  places  may  be  multi- 
plied together.  The  decimal  scale  and  operations 
with  decimals  may  be  studied  with  profit.  Com- 
pound numbers  are  studied  with  reference  to  stand- 
ard units  of  linear,  liquid,  square,  and  dry  measure, 
and  avoirdupois  weight.  Short  division,  and  from 
this  process,  long  division. 

In  work  on  fractions,  the  use  of  names  and  sym- 
bols, reduction,  and  application,  with  reference  to 
money  systems  and  objects  of  daily  use,  should  be 
stressed. 

Students  should  be  taught  care  in  statement  of 
problems,  use  of  words,  and  should  be  kept  busy 
upon  applications  to  home   duties,   and   to   other 
subjects  of  study. 
Summary  : 

Addition  may  be  extended  to  include  numbers 
as  high  as  ten  places  or  more,  if  conditions  warrant. 
Applications  should  be  made  through  concrete  ex- 
amples, with  especial  drill  upon  the  common  things 
of  life,  as  for  instance,  the  money  system. 

Subtraction  of  numbers,  involving  borrowing 
two  or  three  place  numbers,  may  be  used. 

Multiplication  of  selected  numbers  to  three  places, 
*and    drill   upon    the    multiplication    table.     When 
occasion  arises  introductory  work  on  factoring  may 
be  carried  on. 


STATE  N 


50  MATHEMATICS  AND  EDUCATION 

Division,  as  the  inverse  of  multiplication,  and 
work  in  short  and  long  division. 

The  work  in  fractions  may  be  extended  to  cover 
addition  and  subtraction  of  like  fractions,  and  ad- 
dition and  subtraction  of  mixed  numbers. 

Denominate  numbers  are  to  be  used  and  the 
operations  of  addition  and  subtraction  performed 
upon  them. 

The  Roman  number  system  should  be  extended 
to  enable  students  to  count. 
Fifth  grade 

'  During  the  first  part  of  the  year  the  stress  should 
be  on  the  operations  of  addition,  subtraction,  multi- 
plication, and  division.  Counting,  together  with 
the  reading  and  writing  of  numbers,  should  be 
carried  into  the  large  numbers.  In  the  Roman 
system  the  work  should  be  carried  to  a  point  where 
students  find  no  difficulty  in  reading  and  writing 
numbers.  During  the  last  part  of  the  year,  intro- 
ductory measurements  and  the  solution  of  rather 
simple  problems  in  getting  areas  and  volumes  may 
be  taken  up. 
Summarized,  the  work  covers  material  as  follows: 

1.  Counting,  reading,  and  writing,  both  Arabic 
and  Roman  numbers. 

2.  Four    fundamental    operations.     (Very    im- 
portant.) 

3.  Fractions. 

Application  of  four  fundamental  operations. 
Rigid  reductions. 

Operations  involving  fractions  and  integers 
are  taken  up. 


ARITHMETIC   BY  GRADES  51 

4.  Factoring    with   reference   to   prime   factors, 
and    simple    common    denominators    and    common 
multiples. 

5.  Decimal  fractions. 

Relationship  between  the  common  and  deci- 
mal fraction. 
Application    of    fundamental    operations. 

6.  Areas  and  volumes.     (Like  and  unlike  units 
of   measurement.) 

7.  Denominate    numbers.     (Four    fundamental 
operations.) 

Sixth  grade 

The  work  in  this  grade  takes  on  a  practical  turn, 
and  measurement  and  applications  are  stressed. 

Review  of  fractions,  denominate  numbers,  and 
decimals  should  be  given  early  in  the  year. 

Factoring  with  work  on  Highest  Common  Factor 
and  Lowest  Common  Multiple  may  be  continued. 

In  work  on  measurement  some  educators  prefer 
to  spend  an  entire  semester.  While  this  is  perhaps 
not  best,  it  is  true  that  much  time  can  be  profitably 
spent  on  such  work. 

Care  should  be  exercised  constantly  through  this 
period  to  keep  students  speaking  correctly.  With 
the  giving  of  more  concrete  problems  the  necessity 
for  correct  speaking  is  made  a  more  important 
part  of  the  work.  Especially  should  care  be  exer- 
cised in  the  use  of  new  words  incident  to  the  develop- 
ment of  the  subject. 
Seventh  grade 

The  dominant  thing  during  this  year's  work  is 
the  varied  applications  of  mathematics  to  the  af- 


52  MATHEMATICS  AND  EDUCATION 

fairs  of  life.  Two  phases  stand  out,  the  mechanical 
application  and  the  business  application.  Since 
business  has  an  interest  for  all  students  to  a  greater 
or  less  degree,  while  mechanical  applications  direct- 
ly affect  only  a  part  of  the  student  body,  much  time 
should  be  spent  on  the  basic  principles  of  business 
processes.  This  work  should  be  preceded  by  a 
careful  review  of  common  and  decimal  fractions, 
denominate  numbers,  and  the  completion  of  meas- 
urement, etc. 

As  a  large  topic,  percentage  is  the  point  of  con- 
centration. The  principles  of  percentage  should 
be  emphasized,  and  later,  the  applications,  as  in- 
terest, business  method  (including  profit  and  loss), 
insurance,  taxes,  and  commission  and  discount. 

There  should  also  be  introduced  in  this  grade 
elementary  work  in  the  solution  of  problems  using 
a  symbol  for  the  unknown  number,  and  the  equa- 
tion. This  will  give  the  students  an  added  interest 
in  the  harder  problems  and  open  up  a  method  of 
solution  at  once  more  effective  and  more  interest- 
ing. 

To  the  work  on  measurements  may  be  added  ele- 
mentary work  in  geometric  proofs.     The  work  must 
aim  primarily  toward  a  method  of  proof  and  the 
acquisition  of  simple  facts. 
Eighth  grade 

The  work  of  this  grade  should  be  a  complete  re- 
view of  all  that  has  preceded,  with  special  review 
upon  all  usable  facts.  It  is  at  the  end  of  this  year 
that  many  students  leave  school  or,  what  is  equiva- 


ALGEBRA  53 

lent,  the  first  part  of  the  ninth  grade.  Students 
should  have  covered  the  principles  of  the  subject 
by  this  time  so  that  concrete  problems  may  be  given 
covering  every  part  of  the  subject. 

During  this  year  the  mathematics  should  have 
two  purposes;  (1)  to  give  all  useful  facts  to  the 
students  in  review,  (2)  to  lead  gradually  into  alge- 
bra and  geometry.  This  last  purpose  is  really  very 
important,  owing  to  the  fact  that  many  students 
£,et  discouraged  during  the  first  year  of  the  high 
school  course,  due  to  the  absolute  change  in  subject 
matter  and  methods  of  teaching.  There  has  been 
too  much  of  a  chasm  between  the  graded  school 
and  the  high  school,  and  too  much  made  of  the  need 
of  a  changed  method  of  instruction.  There  has 
been  a  tendency  to  force  students  out  of  school  at 
the  end  of  eight  years  who  could  easily  remain  for 
two  extra  years.  The  development  should  be  as 
gradual  and  natural  as  that  from  the  seventh  to  the 
eighth  grades. 

Algebra 

The  problem  of  algebra  in  the  high  school  is  one 
that  has  been  greatly  discussed  during  the  past 
few  years.  The  movement  has  been  toward  the 
making  of  algebra  practical.  Text -books  are  on 
the  market  stressing  practical  application  more 
than  all  else.  This  movement  is  in  the  right  direc- 
tion but  there  must  be  more  than  practical  applica- 
tion to  justify  a  place  in  the  high  school  course  for 
algebra.  As  a  practical  science  alone,  that  is, 
practical  in  the  sense  of  applying  to  the  small  affairs 


54  MATHEMATICS  AND  EDUCATION 

of  life,  it  is  not  worth  a  year's  time.  Judged  upon 
this  ground  educators  have  good  reason  to  contem- 
plate the  removal  of  the  course  from  the  curriculum. 
Algebra,  however,  has  values  far  and  above  this 
sort  of  a  practical  one.  It  is  absolutely  essential 
to  all  who  contemplate  work  in  advanced  science 
or  in  higher  mathematics.  As  a  training  in  general- 
ization it  is  unequaled.  Either  of  these  values 
alone  would  justify  the  retention  of  the  subject. 
In  the  seventh  or  eighth  grades  the  use  of 
the  literal  symbol  should  be  introduced  as  a  symbol 
for  the  unknown  number  in  the  problem,  thus  mak- 
ing the  equation  a  useful  tool  in  the  solution  of  the 
problem.  This  will  not  only  add  interest  and  tend 
toward  greater  efficiency  but  will  open  the  student's 
mind  to  the  next  step  in  the  development  of  number 
conception,  that  of  generalization.  To  follow  this 
introduction  with  a  course  in  algebra  presented  as 
a  generalization  of  arithmetic  solves  the  problem  of 
spanning  the  gap  at  present  existing  between  arithme- 
tic and  algebra.  Always  there  must  be  a  vital 

connection  with  the  processes  of  arithmetic. 
•"  i 

Geometry 

The  values  of  geometry  are,  (1)  the  body  of  facts 
relating  to  measurement  of  plane  and  solid  figures 
and,  (2)  the  training  in  methods  of  proof.  The 
selection  of  theorems  to  be  proved  should  be  made 
upon  this  basis,  that  the  number  of  theorems  proved 
is  not  so  essential  as  the  careful  discussion  of  a  few 
basic  ones. 


GEOMETRY  55 

There  is  too  much  of  text-book  teaching,  the 
memorizing  of  proved  theorems,  and  too  little  of 
original  thinking.  Merely  to  reflect  what  is  found 
in  the  book  gives  very  little  development  and  is 
a  course  in  memory  training  rather  than  a  course 
in  reasoning.  The  value  of  the  science  to  the  pupil 
is  in  the  stimulation  of  the  habit  of  carrying  an 
argument  from  accepted  truths  to  a  logical  con- 
clusion. Except  in  the  case  of  a  few  theorems 
the  valuable  thing  is  not  the  conclusion  but  rather 
the  development  of  the  power  to  reach  it. 

Teachers  would  do  well  to  use  the  text  only  as 
a  reference  book  and  spend  nearly  all  the  time  on 
original  problems,  taking  care  that  the  sequence  of 
the  theorem  is  from  the  simple  to  the  difficult. 
The  text  with  model  proofs  is  of  course  necessary 
during  the  elementary  work,  and  the  basic  theorems 
must  be  proved.  By  careful  suggestion  upon  the 
part  of  the  teacher,  the  theorems  presented  in  the 
text  may  be  proved  by  the  student  as  original 
theorems.  Constant  quibbling  over  minor  points 
and  insistance  upon  stereotyped  methods  have  a 
tendency  to  deaden  the  interest. 

Let  the  study  of  geometry  aim  toward  the  life 
of  the  individual.  Being  able  to  reproduce  a  num- 
ber of  theorems  as  given  in  a  text  is  of  little  moment, 
but  clear,  consistent,  reasoning  ability  is  valuable 
to  every  man  and  woman.  The  average  citizen 
very,  very  seldom  has  occasion  to  resort  to  the 
knowledge  of  geometric  theorems,  but  a  dozen  times 
each  day  he  needs  to  solve  some  situation.  Start 


56  MATHEMATICS  AND  EDUCATION 

the  student  thinking  and  the  usable  facts  will  be 
remembered  without  effort. 

Vocational  departments 

The  movements  in  vocational  work  are  (1)  general 
training  of  all  children  in  simple  construction  and 
in  use  of  tools,  (2)  the  teaching  of  the  trades,  (3) 
the  continuation  schools. 

With  respect  to  the  first  movement,  all  of  the 
larger  schools  and  many  of  the  smaller  ones  offer 
some  work  in  the  manual  arts.  While  as  yet  the 
work  is  often  crude  and  inefficient,  the  ideas  of 
manual  training  and  domestic  science  are  pretty 
generally  accepted.  The  problem  is  not  one  of 
introduction  into  the  schools  so  much  as  one  of 
providing  equipment  and  teachers.  This  will  work 
itself  out  rapidly.  Such  work  demands  the  ele- 
ments of  mathematics  and  offers  great  opportunity 
for  teachers  in  the  way  of  applied  problems. 

The  trade-schools  are  usually  conducted  as  such, 
and  are  not  connected  with  the  public  schools. 
However  those  who  enter  the  trade  schools  usually 
get  their  elementary  work  in  the  public  schools  and 
must  depend  upon  the  training  received  in  them 
for  their  number  conception.  In  the  work  of  the 
trade  school  the  mathematics  usually  embodies 
the  facts  and  principles  involved  in  particular 
trades,  as  for  instance,  carpentry,  book-binding, 
salesmanship,  machine  work,  etc.  Each  particular 
trade  has  its  own  body  of  mathematical  truths  but 
all  depend  on  the  basic  principles  of  number.  Here 
again  the  teacher  may  find  excellent  opportunity 


VOCATIONAL  DEPARTMENTS  57 

for  application,  by  taking  in  turn  problems  arising 
in  the  various  trades. 

The  continuation-school,  although  well  established 
in  Germany,  is  a  new  idea  for  American  cities. 
The  students  upon  leaving  school  to  enter  the  var- 
ious lines  of  vocational  work  are  kept  at  school  a 
few  hours  each  week.  They  then  receive  training 
in  the  particular  trade  in  which  they  are  interested. 
It  is  very  apparent  to  these  students  that  education 
means  dollars  and  cents  to  them  and  that  the  work 
they  had  disliked  before  now  becomes  of  the  great- 
est interest  to  them.  The  active  teacher  will  at- 
tempt to  lay  the  foundations  of  number  before  all 
students  with  a  direct  reference  to  the  working  life 
of  the  adult.  For  instance  a  problem  in  percentage 
means  much  more  to  a  student  when  the  student 
is  determining  the  amount  of  money  he  or  she 
might  receive  at  a  rate  of  2%  on  the  day's  sales. 
Point  out  that  clerks  often  get  part  of  their  salaries 
in  this  way  and  for  a  moment  allow  the  student  to 
pretend  to  be  a  salesman. 


Covering  the  materials  of  mathematics  there  can 
be  no  rigid  rule  to  follow  grade  by  grade,  owing  to 
the  great  difference  in  students  and  teachers.  The 
large  idea  to  keep  in  mind  is  that  each  and  every 
student  in  school  is  to  become  a  citizen  of  the  com- 
munity. He  or  she  must  find  an  occupation  and 
take  a  part  in  the  community  life.  It  may  be  to 
fill  the  high  place  or  the  low  place  in  society,  but, 
nevertheless,  to  work  out  an  existence,  together 


58  MATHEMATICS  AND  EDUCATION 

with  a  certain  degree  of  happiness  and  contentment. 
It  is  beyond  the  teacher's  power  to  decide  what  is 
to  be  the  exact  station  of  each  child,  but  this  much 
he  or  she  does  know,  that,  whatever  it  be,  the  future 
man  or  woman  will  bless  those  who  contributed 
toward  the  making  of  a  well  balanced  man  or  woman. 
Teachers  are,  indeed,  guardians  of  youth,  and  very, 
very  much  is  upon  their  shoulders.  Only  the  best 
of  our  young  men  and  women  should  have  the  privi- 
lege of  caring  for  those  who  are  to  be,  in  total,  our 
future  state.  Teaching  is  never  a  task,  but  rather 
a  wonderful  opportunity.  Hold  to  your  ideals, 
teachers,  and  elevate  your  profession. 

Methods  and  Modes 

In  determining  the  method  and  the  mode  of 
teaching  mathematics  the  teacher  must  have  one 
large  fact  constantly  in  view,  namely,  that  more 
than  80%  of  students  in  school  will  not  enter  the 
high  school.  This  means  that  over  four-fifths  of 
the  students  in  school  must  get  all  the  school  train- 
ing in  the  grades  below  the  ninth.  It  follows  that 
if  the  teacher  plans  the  mathematics  as  a  prepara- 
tion of  higher  subjects  the  whole  effort  is  in  vain 
for  over  four-fifths  of  the  students,  except  for  the 
values  they  get  incidentally. 

If  the  higher  branches  are  worth  sacrificing  four- 
fifths  of  the  student  body  for,  would  it  not  be  better 
to  reconstruct  in  such  a  way  that  the  energy  of 
preparation  is  saved,  and  to  give  in  its  place  the 
elementary  parts  of  the  advanced  branches?  Alge- 
bra and  geometry  are  subject  to  unending  criticism 


METHODS  AND  MODES  59 

because  of  their  impracticability.  The  truth  of  the 
matter  is  that  there  is  very  much  of  value  to  all 
students  in  algebra  and  geometry,  and  a  great  deal 
that  is  not  valuable  to  all  students  in  the  courses  in 
arithmetic  as  offered.  Courses  in  trigonometry, 
analytic  geometry,  and  calculus  also  have  valuable 
facts  for  students  to  know.  Further,  these  advanced 
courses  offer  as  good  training  for  the  mental  faculties 
as  excessive  arithmetic,  and  are  as  easily  grasped  in 
their  elementary  forms  by  young  students  as  arith- 
metic. There  is  only  one  condition  when  mutual 
exclusiveness  holds  good,  that  being  when  all  stu- 
dents complete  the  entire  course. 

Frankly,  then,  would  it  not  be  better  for  us  to 
readjust  so  that  all  students  may  have  the  oppor- 
tunity to  get  an  elementary  knowledge  of  the  sub- 
jects at  present  kept  from  them  ?  They  would  lose 
nothing  in  mental  training.  They  would  gain  in 
the  knowledge  of  facts.  About  all  the  students 
would  lose  is  some  of  the  hatred  they  now  have  for 
eternal  drill  upon  uninteresting  parts  of  arithmetic. 

The  parts  of  algebra  of  value  to  younger  students 
are  literal  symbolism,  the  equation  (simple  in  one 
and  two  unknown  quantities),  and  an  elementary 
knowledge  of  the  fundamental  operations.  Logar- 
ithms are  very  valuable  for  approximating.  From 
geometry  one  should  use  methods  of  proof,  con- 
struction and  estimates  of  areas  and  volumes.  From 
trigonometry,  methods  of  solution  of  triangles 
should  be  studied.  From  analytic  geometry  and 
calculus,  graphical  representation,  and  methods  of 


60  MATHEMATICS  AND  EDUCATION 

approximation  can  be  considered  with  profit.  All 
this  material  can  be  worked  into  the  mathematics 
of  the  sixth,  seventh  and  eighth  grades  with  no  loss 
whatever  to  the  child,  as  far  as  mental  training  goes, 
and  with  great  profit  as  to  acquisition  of  fact  values. 
Perhaps  there  is  truth  in  the  statement  that 
prospective  teachers  of  mathematics  would  do  well 
to  forget  all  that  they  have  heard  of  method  and 
mode,  thus  giving  room  for  so-called  common  sense. 
In  many  cases  there  can  be  no  doubt  of  its  truth. 
The  moment  the  prospective  teacher  loses  the  con- 
nection between  the  method  or  the  mode  and  its 
purpose,  is  the  moment  forgetting  becomes  a  virtue. 
There  is  but  one  problem  always  and  forever  before 
the  teacher,  to  interest  the  pupil,  to  stimulate  a 
desire  for  more  knowledge  upon  the  subject  at  hand, 
and  to  choose  and  present  such  material  as  will  carry 
the  student  to  the  goal.  There  can  not  be  many 
methods,  and  these  methods  cannot  be  mutually 
exclusive.  There  are  different  methods  which, 
though  shading  one  into  the  other,  exist  distinctly. 
These  are  the  synthetic,  the  analytic,  the  deductive, 
the  inductive,  the  Socratic,  the  Heuristic,  and  the 
laboratory  methods.  While  the  teacher  never  finds 
it  possible  to  employ  these  methods  individually 
in  practice,  owing  to  his  personal  temperament  and 
the  unique  reaction  of  his  own  student  group,  yet 
there  is  advantage  for  him  to  appreciate  the  classi- 
fication and  the  advantages  of  each  particular 
method.  It  will  aid  him  in  the  construction  of  his 
working  machinery,  the  machinery  of  the  recitation 


ANALYSIS  AND  SYNTHESIS  61 

the  machinery  of  his  mode  of  presentation.  As 
one  whose  aim  is  to  aid  others  to  acquire  knowledge 
of  a  subject,  he  saves  time  and  energy  by  and  through 
his  application  of  method  in  the  selection  and  ar- 
rangement of  material.  Each  method  has  its  ad- 
vantages in  the  adaptation  to  special  situations. 

The  synthetic  method  is  usually  thought  of  in 
connection  with  the  analytic  method.  It  is  a  mat- 
ter of  common  belief  that  the  one  puts  together 
while  the  other  tears  apart.  While  this  is  not 
strictly  true  the  basic  principle  is  valid.  The  syn- 
thetic method  arrives  at  the  desired  conclusion  by 
application  of  known  truths.  It  is,  therefore,  de- 
ductive in  its  arrangements,  working  from  the  known 
to  the  unknown.  The  analytic  method  breaks  the 
larger  unsolved  problems  into  its  parts,  thus  pro- 
ducing more  simple  problems  which  are  more  easily 
solved.  It  is,  therefore,  inductive  in  its  arrange- 
ments, working  from  the  particular  to  the  general. 
It  does  not  follow,  however,  that  the  synthetic 
method  is  identical  with  the  deductive  method,  or 
that  the  analytic  method  is  identical  with  the  in- 
ductive method.  There  is  a  shading  between  the 
two  in  each  case. 

It  follows  that  the  synthetic  method  is  the  one 
for  the  elegant  proof  and  the  analytic  method  is 
the  one  for  the  workman  searching  for  facts  in  the 
problem.  In  the  class  room  the  synthetic  method 
is  not  the  best  for  daily  use,  owing  to  the  questions 
which  arise  in  the  student's  mind  as  to  why  the 
particular  operations  were  indulged  in  rather  than 


62  MATHEMATICS  AND  EDUCATION 

others.  There  is  proof  with  no  explanation  other 
than  that  it  serves  the  purpose  at  hand.  The  ana- 
lytic method  is  the  one  to  stimulate  the  student  to 
attack  the  problem,  to  tear  it  into  its  parts,  to  find 
the  truth  or  the  known  and  thus  to  solve  the  prob- 
lem. Once  solved,  the  synthetic  method  may  be 
applied  with  profit  and  pleasure,  for  now  the  stu- 
dent appreciates  the  beauty  and  advantage  of  the 
clear  and  direct  line  of  reasoning  leading  from  the 
given  or  admitted  part  of  the  statement  to  the  de- 
sired conclusion.  A  very  common  example  of  syn- 
thetic method  is  to  be  found  in  the  proof  of  the 
geometric  theorem  in  which  the  proof  proceeds  from 
axioms  and  definitions  to  a  logical  conclusion.  This 
method  of  presenting  geometry  is  giving  way  to 
one  in  which  the  student  is  encouraged  to  search 
out  the  proof  unaided.  The  analytic  method  is 
that  employed  in  much  of  algebra,  for  instance,  in 
the  solution  of  an  equation.  Nearly  all  theorems 
are  proved  by  analytic  methods  originally  by  the 
mathematicians;  therefore  it  is  advisable  to  encour- 
age students  to  work  out  their  own  solutions  rather 
than  to  present  the  finished  work  for  them  to  mem- 
orize. One  fault  of  text  book  teaching  is  that  the 
student  is  forced  to  use  the  synthetic  method,  this 
because  the  synthetic  method  is  that  which  gives 
the  text  its  finish  and  certainty,  and  is  used  by 
authors  for  that  reason. 

There  is  a  happy  medium  to  which  teachers  should 
attempt  to  approximate.  This  medium  would  be 
a  method  by  which  the  class-room  work  would  fol- 


INDUCTION  AND  DEDUCTION  63 

low  the  analytical,  and  the  final  statements  would 
be  reconstructed  and  preserved  in  the  synthetical. 
Absolute  hobby-riding  and  hero  worship  as  to 
method  is  one  thing  teachers  must  ward  against, 
Too  often  a  teacher,  because  of  momentary  success, 
adopts  one  method  exclusively,  or  because  of  the 
signal  success  of  a  brilliant  teacher  of  his  acquain- 
tance, he  adopts  a  certain  method,  hoping  for  like 
success.  Each  teacher  as  well  as  each  class  pos- 
sesses an  individuality,  and  methods  should  be  de- 
vised to  meet  the  requirements  of  the  hour. 

The  commonness  of  the  words,  deductive  and  in- 
ductive, would  seem  to  make  comment  unnecessary. 
However,  contrary  to  common  notions,  mathematics 
in  the  constructive  stage  is  as  inductive  as  deduc- 
tive, although  common  belief  is  that  mathematics 
is  the  deductive  science.  No  doubt  the  widespread 
influence  of  Descartes  has  had  much  to  do  with  the 
prevalent  idea.  The  finished  form  is  deductive, 
but  the  working  method  is  quite  as  inductive  as 
deductive.  The  use  of  the  inductive  method  is 
growing,  just  as  the  use  of  the  analytic  method  is 
growing.  It  is  to  be  hoped  that  thinking  teachers 
will  aid  the  advance  of  induction  without  losing  the 
appreciation  of  the  value  of  the  deductive  method 
for  the  finished  proof.  First  allow  the  student  to 
become  interested  in  particular  problems.  The 
solution  of  these  will  generate  a  desire  to  know  the 
general  rule  and  its  proof.  From  this,  in  turn,  the 
advance  to  the  particular  is  welcomed.  An  example 
might  be  cited  in  the  teaching  of  addition.  First 


64  MATHEMATICS  AND  EDUCATION 

create  interest  by  particular  examples  which  can 
be  visualized.  From  the  special  cases  the  rule  is 
gradually  learned.  The  rule  is  then  applied  to  the 
particular.  Thus  the  student  goes  over  the  ground 
in  its  entirety,  developing  the  rule  and  applying  it. 
The  introduction  of  induction  into  mathematics 
in  no  wise  detracts  from  the  characteristic  of  exact- 
ness which  is  one  of  the  subject's  attractions. 
Rather,  it  eases  the  path  by  which  the  student 
comes  to  the  final  statements.  It  injects  the  feel- 
ing of  the  discoverer  into  the  student,  so  developing 
the  feeling  of  ownership  which  is  so  essential  to  the 
state  of  satisfaction. 

The  Heuristic  method  is  the  method  of  encourag- 
ing the  student  to  discover.  The  teacher  must 
see  to  it  that  the  student  does  not  waste  time  and 
energy  upon  false  leads,  beyond  the  point  of  ap- 
preciation of  their  falsity.  The  method  is  perhaps 
the  most  valuable  for  the  teacher  to  adopt  if  the 
teacher  uses  it  in  the  proper  way.  As  an  absolute 
dictum  it  is  as  unfruitful  of  results  as  any  other 
one  used  absolutely.  There  is  as  little  result  in 
asking  students  to  discover  things  without  sug- 
gestion as  to  ask  students  to  drill  eternally  upon 
prescribed  rules  and  formulas.  Used  with  the 
proper  discretion  it  is  the  common  sense  method  for 
excellence.  It  has  its  advantages  and  disadvan- 
tages, its  merits  and  demerits,  but  on  the  whole  the 
spirit  of  the  method  is  good.  It  stimulates  effort 
if  suggestion  is  properly  given.  It  prepares  the 
way  for  original  thinking,  provided  the  teacher 


THE  LABORATORY  METHOD  65 

makes  suggestions  enough  to  keep  the  students 
interested.  This  method  has  been  tried  and  has 
not  proved  successful  in  the  pure  form.  Too  much 
time  is  necessary  for  one  thing.  Not  only  does  the 
class  recitation  move  slowly,  but  much  outside 
study  is  necessary,  that  is,  to  complete  courses  that 
are  at  present  required.  Again,  there  is  a  great 
tendency  for  the  unscrupulous  teacher  to  indulge 
in  relaxation  of  effort,  hiding  behind  the  method 
which  asks  the  student  to  work  without  help.  This 
is  deadening  for  the  student.  The  student  is  as 
helpless  without  suggestion  as  he  would  be  in  a 
chemical  laboratory.  This  point  becomes  espec- 
ially strong  when  one  considers  that  the  average 
intelligence  of  a  public  school  class  is  low  and  the 
young  students  have  not  reached  the  age  of  reason. 
Still  again,  to  refer  a  student  to  outside  references 
is  equivalent  to  allowing  him  to  take  facts  directly 
from  outside  sources.  This,  except  for  the  gain 
in  prolonged  search,  is  the  same  as  to  take  facts 
from  the  text  in  use.  But  despite  the  disadvan- 
tages, the  spirit  of  the  method  is  good.  Used  care- 
fully it  is  effective. 

The  laboratory  method  is  comparatively  new, 
though  the  principle  is  much  the  same  as  the  one 
which  prompted  Pestalozzi  to  start  his  great  re- 
form. Pestalozzi  re-established  the  method  of  us- 
ing objects  in  teaching  arithmetic  after  a  three 
century  discard,  dating  from  the  introduction  of 
Hindu  symbolism.  The  laboratory  method  is  a 
new  reform,  encouraging  the  use  of  all  available 


66  MATHEMATICS  AND  EDUCATION 

material  in  the  teaching  of  mathematics,  supple- 
mented by  the  cutting  out  of  a  great  part  of  what 
is  now  taught.  More  ground  is  covered  in  less  time 
with  stress  placed  upon  acquisition  of  facts.  The 
method  is  built  upon  the  idea  that  since  interest 
is  the  essential  thing  in  teaching  and  since  facts 
are  the  essential  thing  in  life,  these  facts  should  be 
given  to  the  student  by  contact  with  their  applica- 
tion. This  method  encourages  correlation  of  mathe- 
matics with  the  natural  sciences.  It  suggests  the 
teaching  of  the  same  group  by  the  same  teacher, 
to  give  greater  opportunity  to  the  teacher  for  ap- 
plication. Such  a  method  would  require  a  large 
number  of  axioms  and  accepted  proofs.  It  would 
mean  that  intuition  would  be  used  as  an  aid  to 
reason,  for  nothing  that  was  evident  would  be 
proved.  As  in  the  case  of  the  Heuristic  method, 
there  are  disadvantages  to  be  noted.  Interest  for  the 
child  is  not  identical  with  interest  for  the  adult. 
Often  the  child  can  be  as  easily  interested  in  success- 
ful abstract  operations  as  in  manipulation  of  mechani- 
cal objects.  After  all,  people  enjoy  doing  those 
things  they  are  able  to  do  well.  A  boy  is  as  inter- 
ested in  turning  a  cartwheel  as  he  is  in  laying  brick. 
It  is  not  the  operation  but  the  doing  that  creates 
interest.  The  interest  is  not  something  intrinsic. 
It  is  created  by  the  teacher. 

The  Socratic  method  is  well  known,  the  question 
and  answer  method.  It  is  destructive  and  requires 
excessive  time.  Both  these  facts  are  reasons  why 
it  is  not  a  good  method  to  employ  entirely. 


67 

Teachers  often  become  enthusiastic  over  what 
some  man  has  presented  as  a  model  method,  and 
by  following  it  absolutely  get  into  many  troubles. 
Methods  often  read  well  which  do  not  work  well. 
There  are  many  things  a  teacher  must  constantly 
bear  in  mind.  The  various  methods  are  in  pure 
form  the  constructions  of  men  biased  in  their  opin- 
ions. The  laboratory  method,  for  instance,  in  its 
pure  form  is  as  impractical  as  its  opposite.  The 
Socratic  method  is  an  example  of  a  once  famous 
one  now  in  the  discard,  for  there  was  but  one  Socra- 
tes and  he  will  never  be  reborn.  All  methods  have 
their  merits  and  demerits.  Some  methods,  owing 
to  peculiar  conditions,  are  more  efficient  than  others. 
On  the  whole  the  best  advice  is  that  you  each  de- 
velop your  own.  To  do  this  it  is  needless  to  say 
that  you  will  find  aid  and  inspiration  in  acquiring 
a  knowledge  of  what  other  teachers  believe  and 
what  other  people  do. 

At  the  present  time  the  Montessori  method  is 
claiming  more  attention  than  any  other,  but  it  is 
doubtful  whether  such  a  method  will  be  adopted 
in  the  public  schools  of  this  country  for  some  time 
to  come.  However,  approximations  to  it  can  be 
made  with  profit,  especially  with  regard  to  the 
creation  of  interest  as  a  means  of  solving  the  dis- 
cipline problem  to  replace  the  old  method  of  main- 
taining discipline  of  immobility.  The  Montessori 
school  has  a  setting  and  is  conducted  under  con- 
ditions much  more  favorable  than  is  to  be  found 
in  the  elementary  schools  of  the  United  States,  but 


68  MATHEMATICS  AND  EDUCATION 

the  principles  upon  which  the  method  is  built  are 
so  basic  that  every  teacher  can  apply  them.  In 
brief  the  method  is  one  encouraging  the  spontan- 
eous activity  of  the  child  both  physically  and  men- 
tally, under  direction.  The  school  life  is  made  as 
nearly  like  the  ideal  home  life  as  is  possible.  Play 
is  made  purposeful  without  detracting  from  its 
attractiveness.  Much  of  the  success  of  the  school, 
in  fact  most  of  it,  is  due  to  the  very  remarkable 
woman,  Dr.  Montessori.  Her  ideas  are  not  new 
with  her;  in  fact  they  have  been  the  ideas  of  educa- 
tors for  many  years  and  have  been  worked  upon 
in  many  cases  successfully;  but  it  has  remained 
for  this  woman  to  produce  the  most  striking  suc- 
cesses. 

The  American  schools  are  at  present  conduct- 
ing elementary  work  along  the  same  lines  and 
successfully.  In  our  system  it  is  quite  impossible 
to  produce  the  striking  setting  Dr.  Montessori  has, 
but  it  is  possible  to  make  application  of  the  under- 
lying principles  in  our  elementary  work.  Dr.  Mon- 
tessori follows  Froebel  in  the  theory  of  activity 
but  has  perfected  the  practical  system  to  demon- 
strate the  truth  of  her  belief.  She  is  radical  but 
has  successes  to  back  her  assertions.  Her  system 
is  imperfect,  as  she  admits,  but  it  is  the  greatest 
step  in  the  direction  of  informality  yet  taken. 

The  work  in  numeration  covers  simple  counting, 
the  learning  of  number  names,  the  meaning  of 
zero,  the  fundamental  operations,  and  simple  frac- 
tional ideas.  The  striking  part  of  the  method  is 


INTEREST  INDISPENSABLE  69 

the  appeal  to  the  child's  interest.  The  mode  is 
one  of  arranging  the  games  so  as  to  give  the  child- 
ren the  feeling  that  accuracy  is  the  all  essential 
thing.  For  instance,  one  of  the  first  games  is  to 
hand  the  children  slips  of  paper  upon  which  a  num- 
ber is  written,  zero  included,  and  then  each  of  them 
places  the  slip  of  paper  upon  the  desk,  passes  to  a 
table  and  collects  a  number  of  objects  equal  to  the 
number  written  upon  the  slip.  The  teacher  then 
verifies  the  count.  In  this  game  the  child  drawing 
the  number  zero  remains  quietly  at  the  seat,  not 
taking  objects  from  the  table.  The  teacher  empha- 
sizes the  importance  of  taking  the  exact  number 
of  objects.  The  teaching  of  the  operation  is  car- 
ried on  in  a  similar  way  with  different  games,  but 
advantage  is  always  taken  of  the  child's  natural 
activity  rather  than  repression  of  it. 

Interest  is  essential.  It  does  not  follow  that 
one  must  become  the  slave  of  the  student's  idle 
dreaming.  Systematic  handling  of  the  machinery 
of  the  recitation  is  essential  to  economy  of  time  and 
energy.  It  does  not  follow  that  the  class  room 
work  must  become  purely  formal  and  mechanical. 
Charts  and  mechanical  devices  often  aid  the  student 
to  see  certain  relationships.  It  does  not  follow 
that  the  class  periods  must  be  one  continual  round 
of  manipulation  of  charts,  maps,  blocks,  etc.  Prob- 
lems involving  the  facts  of  farming  stimulate  the 
student  as  do  problems  in  steam  and  electricity. 
It  does  not  follow  that  students  are  never  interested 
in  the  abstract  equation.  Lincoln  became  presi- 


70  MATHEMATICS  AND  EDUCATION 

dent  with  an  educational  foundation  based  upon 
the  three  R's.  It  does  not  follow  that  we  must 
not  consider  other  subjects. 

The  pendulum  swings  from  extreme  to  extreme 
with  teachers  who  become  intoxicated  first  with 
one  man's  theory  then  another's.  The  process  is 
2500  years  old,  yet  the  school-man's  problem  is  as 
great  as  it  was  when  Plato  lectured  in  Athens.  But 
there  has  been  progress.  Perhaps  it  has  been  in 
spite  of  the  school-man.  Perhaps  the  invention 
of  printing,  the  telephone,  the  telegraph,  the  steam 
and  gas  engines,  and  the  modern  newspaper,  have 
done  it.  Be  that  as  it  may,  the  teacher  of  mathe- 
matics is  to  realize  first  of  all  that  mere  aping  the 
work  of  a  genius  will  never  lead  to  success.  You 
must  awake  to  the  fact  that  you  are  a  responsible 
person  about  to  become  adviser  to  a  group  of  young 
men  and  women  who  crave  some  knowledge  of 
number  relation.  Your  tastes  and  hobbies  may 
not  appeal  to  them,  so  do  not  attempt  to  force  your 
ideas  into  their  young  minds.  Use  the  brain  God 
gave  you  and  have  an  ideal  of  success  for  your  goal. 
If  you  are  unfortunate  enough  to  be  without  brains 
or  ideals,  seek  another  occupation. 

The  method  has  to  do  with  the  selction  and  ar- 
rangement of  material;  the  mode  has  to  do  with 
the  presentation  of  it.  It  follows  that  in  great  part 
the  mode  follows  directly  the  method.  In  general, 
however,  the  modes  can  be  roughly  grouped  under 
two  heads — the  lecture  and  the  recitation.  The 
lecture  mode  is  that  in  which  the  teacher  presents 


LECTURE  AND  RECITATION  MODES  71 

all  material,  the  student  taking  notes  upon  the  pre- 
sentation; the  recitation  mode  is  that  which  requires 
the  presentation  to  be  made  by  the  student,  the 
teacher  merely  acting  as  quiz  master.  In  the  ex- 
aggerated form  the  recitation  mode  becomes  an 
examination  mode,  the  teacher  not  even  offering 
suggestions.  About  the  same  criticism  falls  upon 
the  various  modes  that  falls  upon  the  various  meth- 
ods. Each  has  its  advantages  and  disadvantages 
and  each  has  a  place  in  certain  situations.  Proba- 
bly no  one  mode  is  advisable  in  the  pure  form. 

The  mode  of  presentation  should  be  such  a  com- 
bination of  the  Heuristic,  laboratory,  individual, 
and  class  recitation,  as  seems  best  in  the  particular 
school.  Students  must  discover  their  own  truths 
as  far  as  possible,  preferably  by  the  actual  contact 
with  their  applications.  Also,  students  must  have 
opportunity  to  advance  as  rapidly  as  their  ability 
warrants.  Lastly,  there  must  be  class  recitation, 
for  one  large  value  of  school  life  is  the  community 
life,  the  appreciation  of  the  individual's  relation  to 
the  group.  Recitation  in  the  presence  of  others  is 
invaluable  as  a  training  in  a  life  problem — social 
contact. 

Above  all  else,  teachers  must  realize  that  in- 
terest does  not  reside  in  an  object  or  a  method 
except  through  the  teacher's  suggestion.  The  teach- 
er must  create  interest.  Once  created,  interest 
solves  many  problems,  among  them,  discipline,  lack 
of  attention,  and  aimlessness.  Get  the  student's 
interest  and  almost  anything  can  be  accomplished. 


72  MATHEMATICS  AND  EDUCATION 

The  particular  mode  must  vary  for  every  class 
and  every  teacher,  because  both  classes  and  teachers 
have  their  personalities.  No  universal  formula 
can  be  written  to  cover  all  cases.  A  few  things  are 
essential.  (1)  Each  student  should  keep  a  note 
book  in  which  to  keep  a  permanent  reocrd  of  work 
done.  (2)  Each  student  should  have  pencil,  ruler, 
and  compass  in  order  constantly  to  be  ready  to  visu- 
alize the  conditions  of  a  problem.  (3)  The  class 
room  should  be  as  complete  a  laboratory  as  possible, 
in  order  that  students  may  see  the  application  of 
facts  learned.  (4)  Individual  work  should  be  ex- 
hibited to  stimulate  rivalry.  (5)  Slow  students 
should  have  careful  attention  until  the  teacher  dis- 
covers the  reason  for  slowness,  then  corrective 
teaching.  (6)  Good  students  should  have  every 
door  open  to  them  for  advancement. 

Summary 

Mathematics,  that  oldest  of  sciences,  is  up  for 
discussion.  From  the  earliest  times  to  the  most 
recent,  the  body  of  science  coming  under  the  title 
mathematics  has  grown  and  enlarged  until  in  its 
many  branches  and  ever  growing  detail  it  dominates 
all  knowledge.  Years  of  expansion  under  the  spell 
of  men  of  genius  have  been  followed  by  years  of 
quiet,  only  again  to  be  followed  by  greater  expan- 
sion under  other  men  of  genius  until  now,  after 
many  hundreds  of  years,  with  a  legion  of  great  minds 
sacrificed  to  the  cause,  the  science  stands  impreg- 
nable. For  those  who  have  not  the  opportunity 
to  devote  years  of  preparation  to  the  larger  and 


SUMMARY  73 

deeper  study  of  its  far-reaching  expansion,  it  stands 
as  a  wonderful,  mysterious  secret  of  nature.  For 
those  whose  minds  have  travelled  into  those  remote 
regions,  hardly  explored,  there  is  a  charm  as  stimu- 
lating as  that  in  the  investigation  of  the  astronomer 
into  the  almost  infinite  regions  of  space.  Part  after 
part  has  unfolded  until  from  the  original  trunk  many 
branches  have  grown.  Now  in  its  last  successes 
another  phase  presents  itself,  that  of  presenting  the 
fundamental  principles  to  growing  minds  in  a  way  to 
stimulate  more  expansion  and  greater  successes  on 
the  one  hand,  and  on  the  other  in  a  way  to  give  all 
men  the  advantage  of  the  facts  discovered.  The 
question  of  mathematics  is  everywhere  in  the  air, 
and  unfortunately  some  have  misinterpreted  the 
growing  unrest.  It  is  not  so  much  a  criticism  of 
the  science,  as  such,-  as  it  is  a  demand  for  more 
knowledge  of  it.  Years  of  careless  teaching  have 
resulted  in  breeding  discontent.  It  is  now  for  the 
teacher  to  satisfy  the  demand  by  developing  a  bet- 
ter method  and  a  more  effective  mode. 

The  question  of  teaching  mathematics  is  not  a 
new  one,  by  any  means.  Mathematics  teaching 
has  been  before  educators  as  long  as  teaching  has, 
but  never  before  has  the  whole  teaching  problem 
been  so  prominent  as  today.  Free  schools  and  a 
great  increase  in  the  number  of  students,  together 
with  the  great  advance  in  civilization,  has  developed 
many  new  problems.  In  the  last  analysis  the  situa- 
tion is  reduced  to  one  in  which  the  efficiency  of  the 
teacher  is  the  controlling  thing.  Methods  and  modes 


74  MATHEMATICS  AND  EDUCATION 

are  aids  to  him  who  can  apply  them  to  advantage. 
To  all  others  new  ways  to  do  things  are  confusing. 

To  be  successful  the  teacher  must  have  a  back- 
ground more  staple  than  a  mere  method.  The  ele- 
ments of  this  background  are,  (1)  an  appreciation 
of  the  aim  and  scope  of  education  built  up  not  alone 
upon  the  passing  ideals  of  the  present  civilization, 
but  also  upon  the  developing  ideal  of  human  per- 
fection, (2)  a  clear  and  unbiased  judgment  covering 
the  values  of  mathematics  to  the  student,  (3)  ability 
to  choose  the  materials  of  education  with  regard 
to  these  values,  (4)  a  method  of  arrangement  of 
material,  and  (5)  a  mode  of  presentation.  The 
ideal  of  success  is  one  which  stands  before  all  who 
now  teach  and  who  expect  to  teach,  and  to  approxi- 
mate this  ideal  is  the  innermost  aim  of  each  and 
every  one.  It  is  the  ideal  preached  to  all,  by  those 
whose  business  is  that  of  teacher-making,  and  it  is 
the  ideal  as  well,  for  all  who  aspire  to  higher  things 
in  any  work.  To  attain  it  becomes  a  ruling  passion. 

Together  with  a  background,  the  teacher  must 
possess  strength  of  will  to  work  out  the  problem 
before  him  or  her  regardless  of  obstacles.  Stewart 
Edward  White  tells  a  story  of  a  man  in  the  far  north 
who  received  orders  to  find  and  take  an  offender 
of  the  law  of  the  north.  He  knew  what  hardship 
and  suffering  it  meant  to  pursue  one  as  crafty  as 
himself  through  the  wilderness,  but  he  had  no  fault 
to  find  or  questions  to  ask.  The  next  year  he  re- 
turned. He  had  found  his  man  and  completed  his 
task.  What  the  obstacles  were,  and  what  the  suf- 


SUMMARY  75 

fering  was,  no  one  knew,  but  he  had  completed  the 
work  he  had  attempted  and  had  attained  success. 
You  teachers  may  also  attain  success  if  you  possess 
the  power  of  character  to  carry  you  through  every 
situation.  In  modern  language,  be  alive,  be  keen, 
be  forceful,  be  resourceful,  be  open  to  suggestion, 
be  ready  to  profit  by  the  experience  of  those  who 
have  succeeded.  Above  all,  be  honest  with  your- 
self and  with  the  world.  Teaching,  the  oldest  and 
most  respected  of  professions,  has  a  future  for  you, 
if  you  search  for  it,  as  great  as  it  has  offered  those 
who  have  made  the  civilization  which  we  enjoy. 


000933  315 


LOS  ANGELES 
STATE  NORMAL  SCHOOL 


